m.sc. in mathematical modelling and scienti c computing ... · the questions involve basic aspects...
TRANSCRIPT
MSc in Mathematical Modelling and Scientific Computing
Dissertation Projects
December 2013
Contents
1 Projects with the Industrial Sponsors of the MSc 3
11 Sharp mdash Flow and Solidification in Confined Geometries with IndustrialApplications 3
2 Numerical Analysis Projects 4
21 Chebfun Dissertation Topics 4
22 Multi-Structures and Computing in Mixed Dimensions 5
23 Parallel Computing for ODEsPDEs with Constraints 6
24 Segmentation and Registration of Lung Images 7
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting 8
26 Constraints and Variational Problems in the Closest Point Method 9
27 Topics in Matrix Completion and Dimensionality Reduction for Low RankApproximation 10
28 Optimisation of Tidal Turbines for Renewable Energy 11
29 Uncertainty Quantification in Glaciological Inverse Problems 13
210 Edge Source Modelling for Diffraction by Impedance Wedges 14
211 What to do with DLA 15
212 Random Plane Wave and Percolation 16
213 Numerical Solution of Equations in Biochemistry 18
214 Numerical Solution of the Rotating Disc Electrode Problem 19
3 Biological and Medical Application Projects 21
31 Circadian Rhythms and their Robustness to Noise 21
32 The Analysis of Low Dimensional Plankton Models 22
33 Individual and Population-Level Models for Cell Biology Processes 23
1
34 A New Model for the Establishment of Morphogen Gradients 24
35 Modelling the Regrowth and Homoeostasis of Skin 26
36 Modelling the Growth of Tumour Spheroids 27
37 DiscreteHybrid Modelling of Lymphangiogenesis 28
38 Mathematical Modelling of the Negative Selection of T Cells in the Thymus 29
39 The Dynamics and Mechanics of The Eukaryotic Axoneme 30
4 Physical Application Projects 32
41 Swarm Robotics From Experiments to Mathematical Models 32
42 A Simple Model for Dansgaard-Oeschger Events 33
43 Modelling Snow and Ice Melt 34
44 A Network-Based Computational Approach to Erosion Modelling 35
45 Retracting Rims 35
46 Modelling Spray Deposition for Applications in Manufacturing Superca-pacitors 36
47 Mathematical Modelling of Membrane Fouling for Water Filtration 37
48 Flow-Induced ldquoSnap-Throughrdquo 38
49 Plumes with Buoyancy Reversal 39
410 Dislocation Structures in Microcantilevers 40
411 Pattern Formation in Axisymmetric Viscous Gravity Currents Flowingover a Porous Medium 41
412 Finger Rafting The role of Spatial Inhomogeneity in Pattern Formationin Elastic Instabilities 42
5 Networks 44
51 Multilayer Networks 44
52 Computational Topology for Neuroscience 45
2
1 Projects with the Industrial Sponsors of the MSc
11 Sharp mdash Flow and Solidification in Confined Geometries with In-dustrial Applications
Supervisor Prof John WettlauferIndustrial Collaborator Philip RobertsContact wettlaufermathsoxacuk
In a recent Applied Mathematics Industrial Workshop Philip Roberts from Sharp pre-sented a class of problems motivated by a device to be used in water purification Thedevice consists of a series of channels in a metallic mass through which water is flowingThe problem is that the water can freeze too quickly and stop subsequent flow Thishas a profound influence on the efficacy and long-term design issues for the company
The questions involve basic aspects of moving boundaries fluid flow and solidificationand geometry There is a class of questions that can be addressed using the theoreticaledifice of these topics suitably modified for the relevant geometry The project willinvolve development of a mathematical model that addresses the role of time dependencein the thermal boundary conditions for the inflow the role of interfacial kinetics and theimportance of crystallinity in controlling the purification properties The mathematicalmethodology of moving phase boundaries will be modified to consider the specificityof the design problem and they will be tested with experimental measurements in theMathematical Observatory
An expected outcome includes a new working model to provide the basis for furthercollaboration with Sharp
References
[1] J A Neufeld and JS Wettlaufer Shear flow phase change and matched asymptoticexpansions pattern formation in mushy layers Physica D 240 140 2011
[2] M G Worster Solidification of fluids in Perspectives in Fluid Dynamics A Collec-tive Introduction to Current Research Cambridge University Press 2000 pp 393ndash446
3
2 Numerical Analysis Projects
21 Chebfun Dissertation Topics
Supervisor Prof Nick Trefethen in collaboration with other members ofthe Chebfun teamContact trefethenmathsoxacuk
Chebfun is an algorithms and software project based on the idea of overloading Mat-labrsquos vectors and matrices to functions and operators We like to think that ldquoCheb-fun can do almost anything in 1Drdquo (integration optimization rootfinding differen-tial equations) and recently a good deal of it has been extended to 2D too Seehttpwwwmathsoxacukchebfun especially the Guide and Examples
A number of MSc dissertations related to Chebfun have been written in recent yearsThere are many possibilities and we can tailor the project to the studentrsquos interests andexpertise
Here are three specific possibilities with the flavours of 2D computing quadrature andclassic approximation theory
1 Numerical vector calculus The Chebfun2 extension to 2D has made it possiblefor us to compute numerically with ldquodiv grad curl and all thatrdquo These operationscan even be then mapped to 2D surfaces in 3D (seehttpwwwmathsoxacukchebfunexamplesgeomhtmlVolumeOfHeartshtml)Next to nothing has been done to utilize these capabilities so far and there aremany possibilities to explore
2 Computation in inner product spaces Chebfun computes inner productsin the vanilla-flavoured way with (f g) defined as the integral of f(x)g(x) overtheir interval of definition Yet it has the quadrature capabilities to handle otherweight functions such as Chebyshev Gauss-Jacobi or Gegenbauer weights In factChebfun even includes delta functions making computation of Stieltjes integralspossible It would be very interesting to explore building these notions into aChebfun ldquodomainrdquo class so that machine-precision computation in nonstandardinner products could be automated and exploited
3 The Remez algorithm for rational best approximation Chebfunrsquos existingREMEZ command works well for polynomial approximations but for rationalapproximations it is very fragile Can it be improved
The Chebfun team consists of about 8-10 people and an MSc student doing a projectin this area would be welcome to participate in our weekly team meetings Ideally astudent wishing to do a Chebfun-related thesis should have taken the ApproximationTheory course in Michaelmas term
4
22 Multi-Structures and Computing in Mixed Dimensions
Supervisor Dr Colin Macdonald (OCCAM)Contact macdonaldmathsoxacuk
Figure 1 Heat equation inmixed dimensions
The Closest Point Method is a recently developed simpletechnique for computing the numerical solution of PDEson general surfaces [24] It is so general that it can com-pute on surfaces where I donrsquot understand the results Forexample it can compute on problems with variable dimen-sion just as easily as a simple sphere In Figure 1 the pigand sphere are connected with a one dimensional filamentand heat flow is solved over the composite domain Butwhat does such a calculation mean What is the correct solution to such a problem
Figure 2 A multi-structure from [3]
A presentation by Prof Vladimir Mazrsquoya (Liverpool) introduced meto multi-structures [13] An example of a multi-structure problemwould to be determine the eigenvalues of a bridge consisting of solidstructures coupled to thin cables The aim of this project is to learnabout multi-structure problems and do some calculations (eg heatequation or LaplacendashBeltrami eigenvalues) using the Closest PointMethod There are also asymptotic techniques that can be appliedhere letting ε represent the ldquoradiusrdquo of the one-dimensional parts(eg [3])
A reasonable achievement would be showing that the Closest PointMethod computes a solution which is consistent with an asymptoticanalysis for some mixed-dimension multi-structures Or maybe it isnot consistent that would be equally interesting
References
[1] V Kozlov V G Mazrsquoya and A B Movchan Asymptotic analysis of fields inmulti-structures Oxford University Press 1999
[2] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[3] A B Movchan Multi-structures asymptotic analysis and singular perturbationproblems European Journal of MechanicsA Solids 25(4)677ndash694 2006
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
5
23 Parallel Computing for ODEsPDEs with Constraints
Supervisor Dr Colin Macdonald (OCCAM)Collaborators Prof Raymond Spiteri (Saskatchewan) Prof Ping Lin(Dundee)Contact macdonaldmathsoxacuk
b b b b
b b b b
b b b b
b
bc
bc
bc
bc
prediction
correction (l = 1)
correction (l = 2)
correction (l = 3)
tmminus3 tmminus2 tmminus1 tm tm+1
Figure 3 Each row of computationshappens in parallel
This project proposes a parallel time-steppingroutine for differential equation with constraintsThe incompressible NavierndashStokes are an exam-ple of constrained PDEs where the divergencefree condition (the constraint) is enforced by thepressure [3] A multicore idea for ODEs and time-dependent PDEs was developed in [2] where par-allelism was exploited to obtain higher-order ac-curacy Here we propose to exploit parallelism toimpose the constraint in an iterative fashion where all iterations happen in parallel
The project would begin with a brief review of differential-algebraic equations (DAEs)which are a framework for dealing with differential equations with constraints The pro-posed numerical algorithm is the Sequential Regularization Method [1] An implemen-tation in OpenMP Python (using the multiprocessing module) or perhaps Matlabwould be programmed Applications would include multi-body systems (eg the pendu-lum and slider-crank mechanisms) and incompressible NavierndashStokes and these wouldform the test cases
The anticipated achievements include improving a DAE solver a projection-free incom-pressible fluid solver and experience in multicore and parallel computing
References
[1] U Ascher and P Lin Sequential regularization methods for nonlinear higher-indexDAEs SIAM J Sci Comput 18(1)160ndash181 1997
[2] A Christlieb C B Macdonald and B Ong Parallel high-order integrators SIAMJ Sci Comput 32(2)818ndash835 2010
[3] P Lin A sequential regularization method for time-dependent incompressible NavierndashStokes equations SIAM J Numer Anal 34(3)1051ndash1071 1997
[4] C B Macdonald and R J Spiteri The predicted sequential regularization method fordifferential-algebraic equations In C DrsquoAttellis V Kluev and N Mastorakis editorsMathematics and Simulation with Biological Economic and Musicoacoustical Applica-tions pages 107ndash112 WSES Press 2001
6
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
34 A New Model for the Establishment of Morphogen Gradients 24
35 Modelling the Regrowth and Homoeostasis of Skin 26
36 Modelling the Growth of Tumour Spheroids 27
37 DiscreteHybrid Modelling of Lymphangiogenesis 28
38 Mathematical Modelling of the Negative Selection of T Cells in the Thymus 29
39 The Dynamics and Mechanics of The Eukaryotic Axoneme 30
4 Physical Application Projects 32
41 Swarm Robotics From Experiments to Mathematical Models 32
42 A Simple Model for Dansgaard-Oeschger Events 33
43 Modelling Snow and Ice Melt 34
44 A Network-Based Computational Approach to Erosion Modelling 35
45 Retracting Rims 35
46 Modelling Spray Deposition for Applications in Manufacturing Superca-pacitors 36
47 Mathematical Modelling of Membrane Fouling for Water Filtration 37
48 Flow-Induced ldquoSnap-Throughrdquo 38
49 Plumes with Buoyancy Reversal 39
410 Dislocation Structures in Microcantilevers 40
411 Pattern Formation in Axisymmetric Viscous Gravity Currents Flowingover a Porous Medium 41
412 Finger Rafting The role of Spatial Inhomogeneity in Pattern Formationin Elastic Instabilities 42
5 Networks 44
51 Multilayer Networks 44
52 Computational Topology for Neuroscience 45
2
1 Projects with the Industrial Sponsors of the MSc
11 Sharp mdash Flow and Solidification in Confined Geometries with In-dustrial Applications
Supervisor Prof John WettlauferIndustrial Collaborator Philip RobertsContact wettlaufermathsoxacuk
In a recent Applied Mathematics Industrial Workshop Philip Roberts from Sharp pre-sented a class of problems motivated by a device to be used in water purification Thedevice consists of a series of channels in a metallic mass through which water is flowingThe problem is that the water can freeze too quickly and stop subsequent flow Thishas a profound influence on the efficacy and long-term design issues for the company
The questions involve basic aspects of moving boundaries fluid flow and solidificationand geometry There is a class of questions that can be addressed using the theoreticaledifice of these topics suitably modified for the relevant geometry The project willinvolve development of a mathematical model that addresses the role of time dependencein the thermal boundary conditions for the inflow the role of interfacial kinetics and theimportance of crystallinity in controlling the purification properties The mathematicalmethodology of moving phase boundaries will be modified to consider the specificityof the design problem and they will be tested with experimental measurements in theMathematical Observatory
An expected outcome includes a new working model to provide the basis for furthercollaboration with Sharp
References
[1] J A Neufeld and JS Wettlaufer Shear flow phase change and matched asymptoticexpansions pattern formation in mushy layers Physica D 240 140 2011
[2] M G Worster Solidification of fluids in Perspectives in Fluid Dynamics A Collec-tive Introduction to Current Research Cambridge University Press 2000 pp 393ndash446
3
2 Numerical Analysis Projects
21 Chebfun Dissertation Topics
Supervisor Prof Nick Trefethen in collaboration with other members ofthe Chebfun teamContact trefethenmathsoxacuk
Chebfun is an algorithms and software project based on the idea of overloading Mat-labrsquos vectors and matrices to functions and operators We like to think that ldquoCheb-fun can do almost anything in 1Drdquo (integration optimization rootfinding differen-tial equations) and recently a good deal of it has been extended to 2D too Seehttpwwwmathsoxacukchebfun especially the Guide and Examples
A number of MSc dissertations related to Chebfun have been written in recent yearsThere are many possibilities and we can tailor the project to the studentrsquos interests andexpertise
Here are three specific possibilities with the flavours of 2D computing quadrature andclassic approximation theory
1 Numerical vector calculus The Chebfun2 extension to 2D has made it possiblefor us to compute numerically with ldquodiv grad curl and all thatrdquo These operationscan even be then mapped to 2D surfaces in 3D (seehttpwwwmathsoxacukchebfunexamplesgeomhtmlVolumeOfHeartshtml)Next to nothing has been done to utilize these capabilities so far and there aremany possibilities to explore
2 Computation in inner product spaces Chebfun computes inner productsin the vanilla-flavoured way with (f g) defined as the integral of f(x)g(x) overtheir interval of definition Yet it has the quadrature capabilities to handle otherweight functions such as Chebyshev Gauss-Jacobi or Gegenbauer weights In factChebfun even includes delta functions making computation of Stieltjes integralspossible It would be very interesting to explore building these notions into aChebfun ldquodomainrdquo class so that machine-precision computation in nonstandardinner products could be automated and exploited
3 The Remez algorithm for rational best approximation Chebfunrsquos existingREMEZ command works well for polynomial approximations but for rationalapproximations it is very fragile Can it be improved
The Chebfun team consists of about 8-10 people and an MSc student doing a projectin this area would be welcome to participate in our weekly team meetings Ideally astudent wishing to do a Chebfun-related thesis should have taken the ApproximationTheory course in Michaelmas term
4
22 Multi-Structures and Computing in Mixed Dimensions
Supervisor Dr Colin Macdonald (OCCAM)Contact macdonaldmathsoxacuk
Figure 1 Heat equation inmixed dimensions
The Closest Point Method is a recently developed simpletechnique for computing the numerical solution of PDEson general surfaces [24] It is so general that it can com-pute on surfaces where I donrsquot understand the results Forexample it can compute on problems with variable dimen-sion just as easily as a simple sphere In Figure 1 the pigand sphere are connected with a one dimensional filamentand heat flow is solved over the composite domain Butwhat does such a calculation mean What is the correct solution to such a problem
Figure 2 A multi-structure from [3]
A presentation by Prof Vladimir Mazrsquoya (Liverpool) introduced meto multi-structures [13] An example of a multi-structure problemwould to be determine the eigenvalues of a bridge consisting of solidstructures coupled to thin cables The aim of this project is to learnabout multi-structure problems and do some calculations (eg heatequation or LaplacendashBeltrami eigenvalues) using the Closest PointMethod There are also asymptotic techniques that can be appliedhere letting ε represent the ldquoradiusrdquo of the one-dimensional parts(eg [3])
A reasonable achievement would be showing that the Closest PointMethod computes a solution which is consistent with an asymptoticanalysis for some mixed-dimension multi-structures Or maybe it isnot consistent that would be equally interesting
References
[1] V Kozlov V G Mazrsquoya and A B Movchan Asymptotic analysis of fields inmulti-structures Oxford University Press 1999
[2] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[3] A B Movchan Multi-structures asymptotic analysis and singular perturbationproblems European Journal of MechanicsA Solids 25(4)677ndash694 2006
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
5
23 Parallel Computing for ODEsPDEs with Constraints
Supervisor Dr Colin Macdonald (OCCAM)Collaborators Prof Raymond Spiteri (Saskatchewan) Prof Ping Lin(Dundee)Contact macdonaldmathsoxacuk
b b b b
b b b b
b b b b
b
bc
bc
bc
bc
prediction
correction (l = 1)
correction (l = 2)
correction (l = 3)
tmminus3 tmminus2 tmminus1 tm tm+1
Figure 3 Each row of computationshappens in parallel
This project proposes a parallel time-steppingroutine for differential equation with constraintsThe incompressible NavierndashStokes are an exam-ple of constrained PDEs where the divergencefree condition (the constraint) is enforced by thepressure [3] A multicore idea for ODEs and time-dependent PDEs was developed in [2] where par-allelism was exploited to obtain higher-order ac-curacy Here we propose to exploit parallelism toimpose the constraint in an iterative fashion where all iterations happen in parallel
The project would begin with a brief review of differential-algebraic equations (DAEs)which are a framework for dealing with differential equations with constraints The pro-posed numerical algorithm is the Sequential Regularization Method [1] An implemen-tation in OpenMP Python (using the multiprocessing module) or perhaps Matlabwould be programmed Applications would include multi-body systems (eg the pendu-lum and slider-crank mechanisms) and incompressible NavierndashStokes and these wouldform the test cases
The anticipated achievements include improving a DAE solver a projection-free incom-pressible fluid solver and experience in multicore and parallel computing
References
[1] U Ascher and P Lin Sequential regularization methods for nonlinear higher-indexDAEs SIAM J Sci Comput 18(1)160ndash181 1997
[2] A Christlieb C B Macdonald and B Ong Parallel high-order integrators SIAMJ Sci Comput 32(2)818ndash835 2010
[3] P Lin A sequential regularization method for time-dependent incompressible NavierndashStokes equations SIAM J Numer Anal 34(3)1051ndash1071 1997
[4] C B Macdonald and R J Spiteri The predicted sequential regularization method fordifferential-algebraic equations In C DrsquoAttellis V Kluev and N Mastorakis editorsMathematics and Simulation with Biological Economic and Musicoacoustical Applica-tions pages 107ndash112 WSES Press 2001
6
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
1 Projects with the Industrial Sponsors of the MSc
11 Sharp mdash Flow and Solidification in Confined Geometries with In-dustrial Applications
Supervisor Prof John WettlauferIndustrial Collaborator Philip RobertsContact wettlaufermathsoxacuk
In a recent Applied Mathematics Industrial Workshop Philip Roberts from Sharp pre-sented a class of problems motivated by a device to be used in water purification Thedevice consists of a series of channels in a metallic mass through which water is flowingThe problem is that the water can freeze too quickly and stop subsequent flow Thishas a profound influence on the efficacy and long-term design issues for the company
The questions involve basic aspects of moving boundaries fluid flow and solidificationand geometry There is a class of questions that can be addressed using the theoreticaledifice of these topics suitably modified for the relevant geometry The project willinvolve development of a mathematical model that addresses the role of time dependencein the thermal boundary conditions for the inflow the role of interfacial kinetics and theimportance of crystallinity in controlling the purification properties The mathematicalmethodology of moving phase boundaries will be modified to consider the specificityof the design problem and they will be tested with experimental measurements in theMathematical Observatory
An expected outcome includes a new working model to provide the basis for furthercollaboration with Sharp
References
[1] J A Neufeld and JS Wettlaufer Shear flow phase change and matched asymptoticexpansions pattern formation in mushy layers Physica D 240 140 2011
[2] M G Worster Solidification of fluids in Perspectives in Fluid Dynamics A Collec-tive Introduction to Current Research Cambridge University Press 2000 pp 393ndash446
3
2 Numerical Analysis Projects
21 Chebfun Dissertation Topics
Supervisor Prof Nick Trefethen in collaboration with other members ofthe Chebfun teamContact trefethenmathsoxacuk
Chebfun is an algorithms and software project based on the idea of overloading Mat-labrsquos vectors and matrices to functions and operators We like to think that ldquoCheb-fun can do almost anything in 1Drdquo (integration optimization rootfinding differen-tial equations) and recently a good deal of it has been extended to 2D too Seehttpwwwmathsoxacukchebfun especially the Guide and Examples
A number of MSc dissertations related to Chebfun have been written in recent yearsThere are many possibilities and we can tailor the project to the studentrsquos interests andexpertise
Here are three specific possibilities with the flavours of 2D computing quadrature andclassic approximation theory
1 Numerical vector calculus The Chebfun2 extension to 2D has made it possiblefor us to compute numerically with ldquodiv grad curl and all thatrdquo These operationscan even be then mapped to 2D surfaces in 3D (seehttpwwwmathsoxacukchebfunexamplesgeomhtmlVolumeOfHeartshtml)Next to nothing has been done to utilize these capabilities so far and there aremany possibilities to explore
2 Computation in inner product spaces Chebfun computes inner productsin the vanilla-flavoured way with (f g) defined as the integral of f(x)g(x) overtheir interval of definition Yet it has the quadrature capabilities to handle otherweight functions such as Chebyshev Gauss-Jacobi or Gegenbauer weights In factChebfun even includes delta functions making computation of Stieltjes integralspossible It would be very interesting to explore building these notions into aChebfun ldquodomainrdquo class so that machine-precision computation in nonstandardinner products could be automated and exploited
3 The Remez algorithm for rational best approximation Chebfunrsquos existingREMEZ command works well for polynomial approximations but for rationalapproximations it is very fragile Can it be improved
The Chebfun team consists of about 8-10 people and an MSc student doing a projectin this area would be welcome to participate in our weekly team meetings Ideally astudent wishing to do a Chebfun-related thesis should have taken the ApproximationTheory course in Michaelmas term
4
22 Multi-Structures and Computing in Mixed Dimensions
Supervisor Dr Colin Macdonald (OCCAM)Contact macdonaldmathsoxacuk
Figure 1 Heat equation inmixed dimensions
The Closest Point Method is a recently developed simpletechnique for computing the numerical solution of PDEson general surfaces [24] It is so general that it can com-pute on surfaces where I donrsquot understand the results Forexample it can compute on problems with variable dimen-sion just as easily as a simple sphere In Figure 1 the pigand sphere are connected with a one dimensional filamentand heat flow is solved over the composite domain Butwhat does such a calculation mean What is the correct solution to such a problem
Figure 2 A multi-structure from [3]
A presentation by Prof Vladimir Mazrsquoya (Liverpool) introduced meto multi-structures [13] An example of a multi-structure problemwould to be determine the eigenvalues of a bridge consisting of solidstructures coupled to thin cables The aim of this project is to learnabout multi-structure problems and do some calculations (eg heatequation or LaplacendashBeltrami eigenvalues) using the Closest PointMethod There are also asymptotic techniques that can be appliedhere letting ε represent the ldquoradiusrdquo of the one-dimensional parts(eg [3])
A reasonable achievement would be showing that the Closest PointMethod computes a solution which is consistent with an asymptoticanalysis for some mixed-dimension multi-structures Or maybe it isnot consistent that would be equally interesting
References
[1] V Kozlov V G Mazrsquoya and A B Movchan Asymptotic analysis of fields inmulti-structures Oxford University Press 1999
[2] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[3] A B Movchan Multi-structures asymptotic analysis and singular perturbationproblems European Journal of MechanicsA Solids 25(4)677ndash694 2006
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
5
23 Parallel Computing for ODEsPDEs with Constraints
Supervisor Dr Colin Macdonald (OCCAM)Collaborators Prof Raymond Spiteri (Saskatchewan) Prof Ping Lin(Dundee)Contact macdonaldmathsoxacuk
b b b b
b b b b
b b b b
b
bc
bc
bc
bc
prediction
correction (l = 1)
correction (l = 2)
correction (l = 3)
tmminus3 tmminus2 tmminus1 tm tm+1
Figure 3 Each row of computationshappens in parallel
This project proposes a parallel time-steppingroutine for differential equation with constraintsThe incompressible NavierndashStokes are an exam-ple of constrained PDEs where the divergencefree condition (the constraint) is enforced by thepressure [3] A multicore idea for ODEs and time-dependent PDEs was developed in [2] where par-allelism was exploited to obtain higher-order ac-curacy Here we propose to exploit parallelism toimpose the constraint in an iterative fashion where all iterations happen in parallel
The project would begin with a brief review of differential-algebraic equations (DAEs)which are a framework for dealing with differential equations with constraints The pro-posed numerical algorithm is the Sequential Regularization Method [1] An implemen-tation in OpenMP Python (using the multiprocessing module) or perhaps Matlabwould be programmed Applications would include multi-body systems (eg the pendu-lum and slider-crank mechanisms) and incompressible NavierndashStokes and these wouldform the test cases
The anticipated achievements include improving a DAE solver a projection-free incom-pressible fluid solver and experience in multicore and parallel computing
References
[1] U Ascher and P Lin Sequential regularization methods for nonlinear higher-indexDAEs SIAM J Sci Comput 18(1)160ndash181 1997
[2] A Christlieb C B Macdonald and B Ong Parallel high-order integrators SIAMJ Sci Comput 32(2)818ndash835 2010
[3] P Lin A sequential regularization method for time-dependent incompressible NavierndashStokes equations SIAM J Numer Anal 34(3)1051ndash1071 1997
[4] C B Macdonald and R J Spiteri The predicted sequential regularization method fordifferential-algebraic equations In C DrsquoAttellis V Kluev and N Mastorakis editorsMathematics and Simulation with Biological Economic and Musicoacoustical Applica-tions pages 107ndash112 WSES Press 2001
6
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
2 Numerical Analysis Projects
21 Chebfun Dissertation Topics
Supervisor Prof Nick Trefethen in collaboration with other members ofthe Chebfun teamContact trefethenmathsoxacuk
Chebfun is an algorithms and software project based on the idea of overloading Mat-labrsquos vectors and matrices to functions and operators We like to think that ldquoCheb-fun can do almost anything in 1Drdquo (integration optimization rootfinding differen-tial equations) and recently a good deal of it has been extended to 2D too Seehttpwwwmathsoxacukchebfun especially the Guide and Examples
A number of MSc dissertations related to Chebfun have been written in recent yearsThere are many possibilities and we can tailor the project to the studentrsquos interests andexpertise
Here are three specific possibilities with the flavours of 2D computing quadrature andclassic approximation theory
1 Numerical vector calculus The Chebfun2 extension to 2D has made it possiblefor us to compute numerically with ldquodiv grad curl and all thatrdquo These operationscan even be then mapped to 2D surfaces in 3D (seehttpwwwmathsoxacukchebfunexamplesgeomhtmlVolumeOfHeartshtml)Next to nothing has been done to utilize these capabilities so far and there aremany possibilities to explore
2 Computation in inner product spaces Chebfun computes inner productsin the vanilla-flavoured way with (f g) defined as the integral of f(x)g(x) overtheir interval of definition Yet it has the quadrature capabilities to handle otherweight functions such as Chebyshev Gauss-Jacobi or Gegenbauer weights In factChebfun even includes delta functions making computation of Stieltjes integralspossible It would be very interesting to explore building these notions into aChebfun ldquodomainrdquo class so that machine-precision computation in nonstandardinner products could be automated and exploited
3 The Remez algorithm for rational best approximation Chebfunrsquos existingREMEZ command works well for polynomial approximations but for rationalapproximations it is very fragile Can it be improved
The Chebfun team consists of about 8-10 people and an MSc student doing a projectin this area would be welcome to participate in our weekly team meetings Ideally astudent wishing to do a Chebfun-related thesis should have taken the ApproximationTheory course in Michaelmas term
4
22 Multi-Structures and Computing in Mixed Dimensions
Supervisor Dr Colin Macdonald (OCCAM)Contact macdonaldmathsoxacuk
Figure 1 Heat equation inmixed dimensions
The Closest Point Method is a recently developed simpletechnique for computing the numerical solution of PDEson general surfaces [24] It is so general that it can com-pute on surfaces where I donrsquot understand the results Forexample it can compute on problems with variable dimen-sion just as easily as a simple sphere In Figure 1 the pigand sphere are connected with a one dimensional filamentand heat flow is solved over the composite domain Butwhat does such a calculation mean What is the correct solution to such a problem
Figure 2 A multi-structure from [3]
A presentation by Prof Vladimir Mazrsquoya (Liverpool) introduced meto multi-structures [13] An example of a multi-structure problemwould to be determine the eigenvalues of a bridge consisting of solidstructures coupled to thin cables The aim of this project is to learnabout multi-structure problems and do some calculations (eg heatequation or LaplacendashBeltrami eigenvalues) using the Closest PointMethod There are also asymptotic techniques that can be appliedhere letting ε represent the ldquoradiusrdquo of the one-dimensional parts(eg [3])
A reasonable achievement would be showing that the Closest PointMethod computes a solution which is consistent with an asymptoticanalysis for some mixed-dimension multi-structures Or maybe it isnot consistent that would be equally interesting
References
[1] V Kozlov V G Mazrsquoya and A B Movchan Asymptotic analysis of fields inmulti-structures Oxford University Press 1999
[2] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[3] A B Movchan Multi-structures asymptotic analysis and singular perturbationproblems European Journal of MechanicsA Solids 25(4)677ndash694 2006
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
5
23 Parallel Computing for ODEsPDEs with Constraints
Supervisor Dr Colin Macdonald (OCCAM)Collaborators Prof Raymond Spiteri (Saskatchewan) Prof Ping Lin(Dundee)Contact macdonaldmathsoxacuk
b b b b
b b b b
b b b b
b
bc
bc
bc
bc
prediction
correction (l = 1)
correction (l = 2)
correction (l = 3)
tmminus3 tmminus2 tmminus1 tm tm+1
Figure 3 Each row of computationshappens in parallel
This project proposes a parallel time-steppingroutine for differential equation with constraintsThe incompressible NavierndashStokes are an exam-ple of constrained PDEs where the divergencefree condition (the constraint) is enforced by thepressure [3] A multicore idea for ODEs and time-dependent PDEs was developed in [2] where par-allelism was exploited to obtain higher-order ac-curacy Here we propose to exploit parallelism toimpose the constraint in an iterative fashion where all iterations happen in parallel
The project would begin with a brief review of differential-algebraic equations (DAEs)which are a framework for dealing with differential equations with constraints The pro-posed numerical algorithm is the Sequential Regularization Method [1] An implemen-tation in OpenMP Python (using the multiprocessing module) or perhaps Matlabwould be programmed Applications would include multi-body systems (eg the pendu-lum and slider-crank mechanisms) and incompressible NavierndashStokes and these wouldform the test cases
The anticipated achievements include improving a DAE solver a projection-free incom-pressible fluid solver and experience in multicore and parallel computing
References
[1] U Ascher and P Lin Sequential regularization methods for nonlinear higher-indexDAEs SIAM J Sci Comput 18(1)160ndash181 1997
[2] A Christlieb C B Macdonald and B Ong Parallel high-order integrators SIAMJ Sci Comput 32(2)818ndash835 2010
[3] P Lin A sequential regularization method for time-dependent incompressible NavierndashStokes equations SIAM J Numer Anal 34(3)1051ndash1071 1997
[4] C B Macdonald and R J Spiteri The predicted sequential regularization method fordifferential-algebraic equations In C DrsquoAttellis V Kluev and N Mastorakis editorsMathematics and Simulation with Biological Economic and Musicoacoustical Applica-tions pages 107ndash112 WSES Press 2001
6
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
22 Multi-Structures and Computing in Mixed Dimensions
Supervisor Dr Colin Macdonald (OCCAM)Contact macdonaldmathsoxacuk
Figure 1 Heat equation inmixed dimensions
The Closest Point Method is a recently developed simpletechnique for computing the numerical solution of PDEson general surfaces [24] It is so general that it can com-pute on surfaces where I donrsquot understand the results Forexample it can compute on problems with variable dimen-sion just as easily as a simple sphere In Figure 1 the pigand sphere are connected with a one dimensional filamentand heat flow is solved over the composite domain Butwhat does such a calculation mean What is the correct solution to such a problem
Figure 2 A multi-structure from [3]
A presentation by Prof Vladimir Mazrsquoya (Liverpool) introduced meto multi-structures [13] An example of a multi-structure problemwould to be determine the eigenvalues of a bridge consisting of solidstructures coupled to thin cables The aim of this project is to learnabout multi-structure problems and do some calculations (eg heatequation or LaplacendashBeltrami eigenvalues) using the Closest PointMethod There are also asymptotic techniques that can be appliedhere letting ε represent the ldquoradiusrdquo of the one-dimensional parts(eg [3])
A reasonable achievement would be showing that the Closest PointMethod computes a solution which is consistent with an asymptoticanalysis for some mixed-dimension multi-structures Or maybe it isnot consistent that would be equally interesting
References
[1] V Kozlov V G Mazrsquoya and A B Movchan Asymptotic analysis of fields inmulti-structures Oxford University Press 1999
[2] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[3] A B Movchan Multi-structures asymptotic analysis and singular perturbationproblems European Journal of MechanicsA Solids 25(4)677ndash694 2006
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
5
23 Parallel Computing for ODEsPDEs with Constraints
Supervisor Dr Colin Macdonald (OCCAM)Collaborators Prof Raymond Spiteri (Saskatchewan) Prof Ping Lin(Dundee)Contact macdonaldmathsoxacuk
b b b b
b b b b
b b b b
b
bc
bc
bc
bc
prediction
correction (l = 1)
correction (l = 2)
correction (l = 3)
tmminus3 tmminus2 tmminus1 tm tm+1
Figure 3 Each row of computationshappens in parallel
This project proposes a parallel time-steppingroutine for differential equation with constraintsThe incompressible NavierndashStokes are an exam-ple of constrained PDEs where the divergencefree condition (the constraint) is enforced by thepressure [3] A multicore idea for ODEs and time-dependent PDEs was developed in [2] where par-allelism was exploited to obtain higher-order ac-curacy Here we propose to exploit parallelism toimpose the constraint in an iterative fashion where all iterations happen in parallel
The project would begin with a brief review of differential-algebraic equations (DAEs)which are a framework for dealing with differential equations with constraints The pro-posed numerical algorithm is the Sequential Regularization Method [1] An implemen-tation in OpenMP Python (using the multiprocessing module) or perhaps Matlabwould be programmed Applications would include multi-body systems (eg the pendu-lum and slider-crank mechanisms) and incompressible NavierndashStokes and these wouldform the test cases
The anticipated achievements include improving a DAE solver a projection-free incom-pressible fluid solver and experience in multicore and parallel computing
References
[1] U Ascher and P Lin Sequential regularization methods for nonlinear higher-indexDAEs SIAM J Sci Comput 18(1)160ndash181 1997
[2] A Christlieb C B Macdonald and B Ong Parallel high-order integrators SIAMJ Sci Comput 32(2)818ndash835 2010
[3] P Lin A sequential regularization method for time-dependent incompressible NavierndashStokes equations SIAM J Numer Anal 34(3)1051ndash1071 1997
[4] C B Macdonald and R J Spiteri The predicted sequential regularization method fordifferential-algebraic equations In C DrsquoAttellis V Kluev and N Mastorakis editorsMathematics and Simulation with Biological Economic and Musicoacoustical Applica-tions pages 107ndash112 WSES Press 2001
6
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
23 Parallel Computing for ODEsPDEs with Constraints
Supervisor Dr Colin Macdonald (OCCAM)Collaborators Prof Raymond Spiteri (Saskatchewan) Prof Ping Lin(Dundee)Contact macdonaldmathsoxacuk
b b b b
b b b b
b b b b
b
bc
bc
bc
bc
prediction
correction (l = 1)
correction (l = 2)
correction (l = 3)
tmminus3 tmminus2 tmminus1 tm tm+1
Figure 3 Each row of computationshappens in parallel
This project proposes a parallel time-steppingroutine for differential equation with constraintsThe incompressible NavierndashStokes are an exam-ple of constrained PDEs where the divergencefree condition (the constraint) is enforced by thepressure [3] A multicore idea for ODEs and time-dependent PDEs was developed in [2] where par-allelism was exploited to obtain higher-order ac-curacy Here we propose to exploit parallelism toimpose the constraint in an iterative fashion where all iterations happen in parallel
The project would begin with a brief review of differential-algebraic equations (DAEs)which are a framework for dealing with differential equations with constraints The pro-posed numerical algorithm is the Sequential Regularization Method [1] An implemen-tation in OpenMP Python (using the multiprocessing module) or perhaps Matlabwould be programmed Applications would include multi-body systems (eg the pendu-lum and slider-crank mechanisms) and incompressible NavierndashStokes and these wouldform the test cases
The anticipated achievements include improving a DAE solver a projection-free incom-pressible fluid solver and experience in multicore and parallel computing
References
[1] U Ascher and P Lin Sequential regularization methods for nonlinear higher-indexDAEs SIAM J Sci Comput 18(1)160ndash181 1997
[2] A Christlieb C B Macdonald and B Ong Parallel high-order integrators SIAMJ Sci Comput 32(2)818ndash835 2010
[3] P Lin A sequential regularization method for time-dependent incompressible NavierndashStokes equations SIAM J Numer Anal 34(3)1051ndash1071 1997
[4] C B Macdonald and R J Spiteri The predicted sequential regularization method fordifferential-algebraic equations In C DrsquoAttellis V Kluev and N Mastorakis editorsMathematics and Simulation with Biological Economic and Musicoacoustical Applica-tions pages 107ndash112 WSES Press 2001
6
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
24 Segmentation and Registration of Lung Images
Supervisors Dr Colin Macdonald and Dr Julia Schnabel (Oxford BiomedicalImage Analysis)Contact macdonaldmathsoxacuk
httpwwwibmeoxacukresearchbiomedia
Figure 4 (a) lung scan (b) magnitude ofdisplacement wo slip (c) with slip (e)ndash(f) zoom near sliding boundary
Computer tomography (CT) magnetic res-onance imaging (MRI) and positron emis-sion tomography (PET) result in 3D volumedata containing representations of tissuesand organs such as the lungs Mathemat-ical image processing techniques are crucialto acquiring and analysing this data Com-monly the 3D data must be aligned viasome function which maps it onto anotherdata setmdashthis is known as registration Ex-amples include multimodal imaging (whereboth CT and less harmful but less accuratePET data are recorded simultaneously) Comparing one patient to a normal healthysample or tracking change over time also require registration During each inhaleexhalecycle the lungs experience significant translationslip relative to the torso Until re-cently registration techniques did not explicitly account for this motion and subsequentlygave poor results
This project would begin with a review of image processing and specifically level settechniques [2] We would try to construct a mathematical model for the registrationproblem that treats the surface of the lungs as well as the 3D voxel data The registrationproblem should not penalize for motion parallel to this surface [3 4] Numerically thesurface would be represented implicitly using level-set or closest-point based techniques[1] We would implement our algorithm (in Matlab or Python) and perform experimentsusing both simulated and real data
As part of a brand-new collaboration this project is very much ldquoblue skiesrdquo We hopeto show feasibility of incorporating more ldquoprior knowledgerdquo of the particular problem oflung image segmentation This could lead to improved registration As the BiomedicalImage Analysis lab is motivated by clinical diagnosis therapy planning and image-basedtreatment guidance the long term goal would be to improve the tools used in practice
References
[1] C B Macdonald and S J Ruuth The implicit Closest Point Method for the nu-merical solution of partial differential equations on surfaces SIAM J Sci Comput31(6)4330ndash4350 2009
[2] S Osher and R Fedkiw Level set methods and dynamic implicit surfaces Springer-Verlag 2003
7
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
[3] B Papiez M Heinrich L Risser and J A Schnabel Complex lung motion estimationvia adaptive bilateral filtering of the deformation field Proceedings for the Medical ImageComputing and Computer Assisted Intervention (MICCAI) 2013
[4] L Risser F-X Vialard H Y Baluwala and J A Schnabel Piecewise-diffeomorphicimage registration Application to the motion correction of 3d CT lung images usingsliding conditions Medical Image Analysis 2013
25 Repairing Damaged Volumetric Data using Fast 3D Inpainting
Supervisors Dr Tom Marz and Dr Colin MacdonaldContact maerzmathsoxacuk and macdonaldmathsoxacuk
Figure 5 Before (top)and after (bottom) in-painting
Digital inpainting fills in missing pixels in a damaged imagesuch as a creased photograph It can also be used to manipu-late images as in Figure 5 This is an inverse problem and isusually regularized in some way so that the resulting image ispleasing in the ldquoeye-ball normrdquo Defining the latter is wherethe mathematics gets interesting There are analogous prob-lems in three-dimensions for example removing watermarksor subtitles from video (2D + time) There are also likely ap-plications in medical imaging This project would investigatethe 3D image inpainting problem on voxel data (see for example[1 7])
The BornemannndashMarz inpainting algorithm is a recent and fastimage inpainting technique [4 5] This project would begin byreviewing the mathematics and implementation of this algo-rithm (we have existing software in Matlab and the GIMP toexperiment with) We would then extend the approach to threedimensions The techniques include finite difference methodsfor PDEs fast marching methods structure tensors and somenumerical linear algebra There are plenty of theoretical math-ematical issues too depending on interest
A minimum goal would be to understand the mathematics and extend our software(githubcommaerztominpaintBCT) to 3D We could then look at applications suchas video inpainting Depending on interest we could also look at inpainting of colourdata on triangulated surfaces (using the Closest Point Method [3 2 6]) or investigateapplications in medical imaging
References
[1] M Bertalmıo A L Bertozzi and G Sapiro NavierndashStokes fluid dynamics andimage and video inpainting In Proc of IEEE International Conference on ComputerVision and Pattern Recognition 2001
[2] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-
8
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
versity 2011
[3] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[4] F Bornemann and T Marz Fast image inpainting based on coherence transportJournal of Mathematical Imaging and Vision 28(3)259ndash278 2007
[5] T Marz Image inpainting based on coherence transport with adapted distancefunctions SIAM Journal on Imaging Sciences 4(4)981ndash1000 2011
[6] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[7] K A Patwardhan G Sapiro and M Bertalmıo Video inpainting of occluding andoccluded objects In Proc of IEEE International Conference on Image Processing 2005
26 Constraints and Variational Problems in the Closest Point Method
Supervisor Dr Colin MacdonaldContact macdonaldmathsoxacuk
Figure 6 Reaction-diffusion equations ona red blood cell surface[5]
The Closest Point Method is a recently developed simple tech-nique for computing the numerical solution of PDEs on generalsurfaces [4] The method works by embedding the surface inthree-dimensions an imposing a constraint to keep the solutionconstant in the normal direction Despite the work [5] and vonGlehnrsquos thesis we still have many basic questions Here are acouple which could make for good MSc projects
(a) Can we interpret the constraint as a differential-algebraicequation (DAE) See also my other project on DAEs whichthis could easily tie into If the constrained problem is indeed aDAE what is its index If its not (technically) a DAE can westill use DAE techniques to solve it How would these compareto [5]
(b) How do we deal with variational approaches to surface prob-lems in this constrained closest-point framework This ties into image processing onsurfaces something developed over several Oxford MSc theses [1 2 3] We knowsome things about surface integrals thanks to Tom Marz How do we formulate EulerndashLagrange equations for these constrained expressions
In either case the project would involve RungendashKutta methods finite difference schemessome numerical linear algebra and some geometry A project would involve a mixtureof theory and practical computation
In any of these projects we would hope to improve our understanding of the constrainedproblem and more generally of numerical techniques for solving PDE problems on curvedsurfaces
9
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
References
[1] H Biddle Nonlinear diffusion filtering on surfaces MSc dissertation Oxford Uni-versity 2011
[2] H Biddle I von Glehn C B Macdonald and T Marz A volume-based method fordenoising on curved surfaces 2013 To appear in Proc ICIP13 20th IEEE InternationalConference on Image Processing
[3] E Naden Fully anisotropic diffusion on surfaces and applications in image processingMSc dissertation Oxford University 2013
[4] S J Ruuth and B Merriman A simple embedding method for solving partialdifferential equations on surfaces J Comput Phys 227(3)1943ndash1961 2008
[5] I von Glehn T Marz and C B Macdonald An embedded method-of-lines approachto solving partial differential equations on surfaces 2013 Submitted
27 Topics in Matrix Completion and Dimensionality Reduction forLow Rank Approximation
Supervisor Prof Jared TannerContact tannermathsoxacuk
Matrix completion concerns recovering a matrix from few of its entries For a generalmatrix with independent entries this task is not possible but for matrices with fur-ther structure the intercorrelation of entries may allow the full matrix to be recoveredThe prototypical assumption of structure is low rank in which case algorithms havebeen shown to be able to recover low rank matrices from asymptotically the optimallyfewest number of measurements that is the number of degrees of freedom in the lowrank matrix This is an active area of research including development of fundamentaltheory algorithms and their application from online recommendation systems to imageprocessing
This topic can accommodate a variety of questions ranging from a) developing anunderstanding of fundamental theory such as the embedding constants of inner matrixinner products with low rank matrices b) implementing and benchmarking competingsimple algorithms in a parallel infrastructure such as graphical processing units c) repro-ducing and if possible extending some of the more complex algorithms such as randomprojection divide and conquer algorithms or d) review literature in tensor decomposi-tions and completions These projects involve a good understanding of numerical linearalgebra some familiarity with probability and computer programming
References
[1] Lester Mackey Ameet Talwalker and Michael I Jordon Distributed Matrix Com-pletion and Robust Factorization httparxivorgabs11070789
[2] N Halko P G Martinsson and J A Tropp Finding Structure with Randomness
10
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
Probabilistic Algorithms for Constructing Approximate Matrix Decompositions SIAMReview 53(2)217ndash288 May 2011
[3] Benjamin Recht Maryam Fazel and Pablo A Parrilo Guaranteed Minimum-RankSolutions of Linear Matrix Equations via Nuclear Norm Minimization SIAM Review52(3)471ndash501 August 2010
[4] Raghunandan H Keshavan Andrea Montanari and Sewoong Oh Matrix completionfrom a few entries IEEE Transactions on Information Theory 56(6)2980ndash2998 June2010
28 Optimisation of Tidal Turbines for Renewable Energy
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement Tidal stream turbines extract energy fromthe movement of the tides in much the same way as wind turbines collect energy fromthe wind The UK has abundant renewable marine energy resources which could sup-ply reliable clean electricity support a new local high-technology industrial sector andreduce emissions of carbon emissions The Carbon Trust predicts that the marine re-newables industry could be worth tens of billions of pounds by 2050 and support tensof thousands of UK-based jobs if the UK moves quickly it could become the worldleader in this technology as Denmark has become in wind turbines However beforethis potential can be realised the industry must solve a design problem In order toextract an economically useful amount of energy large arrays (up to several hundred)must be deployed on a given site How should the turbines in an array be placed toextract the maximum possible energy The configuration makes a major difference tothe power extracted and thus to the economic viability of the installation
Description of the approach planned and techniques needed For a given turbineconfiguration (the control) a set of partial differential equations (the nonlinear shallowwater or Navier-Stokes equations) is to be solved for the resulting flow configurationand the power extracted computed (a functional involving the cube of the flow speed)The optimisation problem is to maximise the power extracted subject to the physicalconstraints and that the design is feasible (eg that the turbines satisfy a minimumdistance constraint that they are deployed within the site licensed etc) In a recentpublication [1] I solve this optimisation problem using the adjoint technique whichsolves an auxiliary PDE that propagates causality backwards allowing for the veryefficient computation of the gradient of the power extracted with respect to the turbinelocations These adjoint PDEs are automatically derived from the forward problem usingthe hybrid symbolicalgorithmic differentiation approach presented in [2] With thisadjoint technique optimisation algorithms that rely on first-order derivative informationmay be used such as the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm[3] However without second-order information the use of more powerful optimisationalgorithms such as Newtonrsquos method is precluded
11
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
What yoursquod hope to achieve In this project the research student will extendthe tidal turbine optimisation solver to use (variants of) Newtonrsquos method for PDE-constrained optimisation It is anticipated that this will greatly accelerate the conver-gence of the optimisation algorithm This will rely on incorporating recent (unpublished)advances in the extremely efficient computation of tangent linear and second-order ad-joint solutions which enable the computation of the necessary second-order derivativeinformation very quickly
(a) Satellite image of Stroma Island andCaithness MeyGen Ltd have licensed thissite to deploy a 398MW array of tidal tur-bines
(b) Computational domain with the tur-bine site marked pink
(c) Initial turbine positions (256 turbines) (d) Optimised turbine positions In thisidealised case the optimisation improvedthe farm efficiency by 32
References
[1] S W Funke P E Farrell and M D Piggott Tidal turbine array optimisation usingthe adjoint approach Renewable Energy 63658ndash673 2014
[2] P E Farrell D A Ham S W Funke and M E Rognes Automated derivation ofthe adjoint of high-level transient finite element programs SIAM Journal on ScientificComputing 35(4)C369ndashC393 2013
[3] J Nocedal and S J Wright Numerical Optimization Second Edition SpringerVerlag 2006
12
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
29 Uncertainty Quantification in Glaciological Inverse Problems
Supervisor Dr Patrick FarrellContact farrellpmathsoxacuk
Background and problem statement The response of the worldrsquos ice sheets toa changing environment is a key ingredient in the understanding of past and futureglobal climate change due to their potential for rapid contributions to sea level change[1] However there remain significant gaps in our understanding of the dynamics offast-flowing glacial ice in part because computer models of ice sheets must take asinput physical properties which are unknown or difficult to measure These unknownproperties such as bedrock topography and ice temperature are often spatially variableand hence there are an extremely large number of unknown inputs which affect thepredictions derived from computer simulations A popular technique for determiningthese unknown values is to use available observations such as satellite-derived altimetryand surface velocities to invert for these values ie to find the values such that themodel output best fits the observations However a question remains to what degreeare the estimated values constrained by those observations
Description of the approach planned and techniques needed Practitioners typi-cally take a deterministic approach to model inversion a single point in parameter spaceis sought that best minimises the misfit functional Adopting a Bayesian perspectivethis is equivalent to minimising the negative log of the posterior density However theBayesian approach offers additional insight the covariance of the posterior distributioncan be locally characterised by computing the eigendecomposition of the misfit Hessianevaluated at that minimiser (see eg [2]) This allows for the identification of directionsin parameter space that are well-constrained or poorly-constrained by the available data
What yoursquod hope to achieve The student will implement a simple discretisation ofthe higher-order Blatter-Pattyn ice sheet model [3] and apply it to steady isothermalsimulations of the Greenland ice sheet (see Figure 7) The student will then generatesynthetic observations from known input data and use it in solving the deterministicinverse problem taking care to avoid ldquoinverse crimesrdquo [4] The student will then applymatrix-free eigendecomposition algorithms to characterise the covariance of the posteriordistribution at that misfit minimiser
References
[1] S Solomon D Qin M Manning Z Chen M Marquis K Averyt M Tignor andH L Miller (Editors) Climate Change 2007 The Physical Science Basis CambridgeUniversity Press 2007
[2] W C Thacker The role of the Hessian matrix in fitting models to measurementsJournal of Geophysical Research 94(C5)6177ndash6196 1989
[3] M Perego M Gunzburger and J Burkardt Parallel finite element implementationfor higher-order ice-sheet models Journal of Glaciology 58(207)76ndash88 2012
[4] J Kaipio and E Somersalo Statistical and Computational Inverse Problems Volume
13
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
Figure 7 The velocity solution of the steady isothermal Blatter-Pattyn equations dis-cretised using finite elements on a 10km mesh of the Greenland ice sheet
160 of Applied Mathematical Sciences Springer-Verlag 2004
210 Edge Source Modelling for Diffraction by Impedance Wedges
Supervisor Dr David HewettPossible Collaborator Prof U Peter Svensson NTNU TrondheimContact hewettmathsoxacuk
Wave scattering problems arise in many applications in acoustics electromagnetics andlinear elasticity However exact solutions to the (apparently simple) wave equationsmodelling these processes are rare One special geometry amenable to an exact analysisis the exterior of a wedge For the wedge scattering problem with sound-soft (Dirichlet)or sound-hard (Neumann) boundary conditions Svensson et al [1] have recently shownhow the ldquodiffractedrdquo field component of the exact closed-form solution can be writtenas a superposition of point sources distributed along the diffracting edge As well asbeing appealing from a physical point of view this ldquoedge sourcerdquo formulation has alsobeen used by Svensson and Asheim [2] to develop a new integral equation formulationfor scattering problems which may offer a promising alternative to existing tools suchas the boundary element method
The aim of this project is to investigate whether or not an edge source solution rep-resentation is possible for the problem of diffraction by a wedge on which impedance(absorbing) boundary conditions are imposed The impedance boundary condition ismore realistic for acoustic modelling of many reflecting surfaces but is more challengingto analyse than the Dirichlet and Neumann cases The project will focus on the specialcase of a right-angled impedance wedge for which Rawlins [3] has shown that the exact
14
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
solution can be expressed in terms of the (known) solution for the Dirichlet wedge
A general background in wave propagation would be useful but is not essential Knowl-edge of complex analysis (in particular complex contour integral manipulations) wouldalso be valuable
References
[1] U P Svensson P T Calamia and S Nakanishi Frequency-domain edge diffractionfor finite and infinite edges Acta acustica united with acustica 95(3)568ndash572 2009
[2] U P Svensson and A Asheim An integral equation formulation for the diffractionfrom convex plates and polyhedra Tech Report TW610 KU Leuven 2012
[3] A D Rawlins Diffraction of an E- or H-polarized electromagnetic plane wave by aright-angle wedge with imperfectly conducting faces Q J Mech Appl Math 43(2)161ndash172 1990
211 What to do with DLA
Supervisors Dr Dmitry Belyaev and Dr Alan HammondContact belyaevmathsoxacuk
The ultimate goal is to prove that the dimension of DLA cluster is strictly less than 2(analog of Kestenrsquos theorem stating that the dimension is at least 32) I propose tostudy the development of DLA cluster started from the large disc
To fix scale we fix the size of the particles to be equal to one We start from the discof radius N (ie dense cluster of N2 particles) We would like to study how fast thefractal structure will appear (will it happen in N2 steps or may be faster or slower)
What should be computed
bull We start with the disc of size N (denoted by D0) First of all we should computeDi for i = 1 N2
bull For some of Di we should compute the distribution of harmonic measure (probablyin the form of dimension spectrum) The best choice would be compute thisfor all values of i but this is will demand too much computer time and eachstep introduces very localized change in harmonic measure I propose to find(experimentally) the time step δ such that within this time change of measuredistribution is globally significant but small
There are several things we could do with the data
bull Check how fast the spectrum approaches the spectrum of DLA cluster (ie howfast the smooth structure of the disc will be forgotten)
bull Compare obtained spectra with the spectra computed by Mandelbrot et al
15
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
bull Compute correlations (rate of their decay) of harmonic measure Namely we startwith N sectors such that they all carry 1N of harmonic measure (strong correlations uniform distribution) After some time the distribution should startresembling DLA and correlations should weaken
212 Random Plane Wave and Percolation
Supervisor Dr Dmitry BelyaevContact belyaevmathsoxacuk
The main goal of this project is to explore connections between two important andinteresting physics models plane waves and percolation The first model appears inthe study of quantum systems and other problems related to the eigenfunctions of theLaplacian the second model is used to describe porous media spread of forest fires aswell as many other phenomena
Random plane waves There are many problems in physics that are related to the studyof eigenfunctions of Laplace operator and their zero level sets (For example the sandon a vibrating plate concentrates on the zero level set of a Laplace eigenfunction) Itis conjectured that for a very large class of domains the typical behaviour of a highenergy eigenfunction is the same as that of a random superposition of simple planewaves (RPW) This motivates our interest in RPW and their nodal lines (curves wherethe plane wave is equal to zero)
(a) Random spherical harmonic an ana-logue of RPW on a sphere (figure by ABarnett)
(b) Sand on a vibrating placte (photosfrom MIT Physics TSG site)
There are several ways to think about RPW they lead to different approximations thatcould be used for simulations The simplest way is to say that a random plane wave isa random linear combination of plane waves A standard plane wave in R2 is
ReAekθmiddotz
where A is the amplitude E = k2 is the energy and θ is a unit vector which gives the
16
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
direction of the wave Simple computation shows that plane wave solves the Helmholtzequation ∆f + k2f = 0 (this is the same as to say that f is an eigenfunction of Laplaceoperator with eigenvalue minusk2) Informally we can define RPW as
Re
(limNrarrinfin
1radicN
Nsumn=1
Cneikθnmiddotz
)where Cn are independent random (complex) normal variables and θn are directionsOne can take θn to be equi-distributed θn = e2πinN or to be N independent uniformlydistributed random unit vectors
The problem is that it is not clear in what sense this series converges and how to proveit on the other hand it might be used for some simulations since it only uses very simpletrigonometric functions For rigorous definition is is better to use another formula
F (z) =
infinsumn=minusinfin
Re(CnJn(kr)einθ
)where z = reiθ Cn as before and Jn is the Bessel function of the first kind Check thatthis function is a solution of Helmholtz equation
We want to study the nodal lines (set where F = 0) of the function F namely we areinterested in their behaviour inside of a fixed domain as k rarr infin This is the same asfixing k = 1 and studying nodal lines in expanding domains
Percolation This is one of the most studied models in statistical physics this year alonethere were 12000 papers on the subject Yet there are many aspects of this model thatare not known rigorously and even not well understood numerically There are manyversions of this model but we will need the simplest one edge percolation We fix somegraph possibly infinite the best example is Z2 grid and probability p isin [0 1] Afterthat we independently keep each edge with probability p or remove it with probability1 minus p Alternatively we can think that edges are ldquoopenrdquo or ldquoclosedrdquo with probabilitiesp and 1minus p We are interested in connected components of this random graph and howtheir structure depend on p
Connection between models It was conjectured that the nodal domains of the randomplane wave are well described by percolation on the square lattice Z2 with p = 12Recent careful numerical experiments have shown that this is not quite true and themodel should be corrected I propose to study percolation on a random graph which isgenerated by RPW in the following way its vertices are given by all local maxima of theplane wave and edges are encoded by the saddle points For each saddle there are twodirections of steepest ascent (following gradient flow lines) that terminate at two localmaxima The edge corresponding to the saddle connects two vertices that correspondto these two local maxima I believe that the percolation with p = 12 on this lattice isa good model for nodal domains
The main parts of the project will be
bull Sample RPW on a relatively large domain The size of the domain should be ofthe order 100λ where λ = 2πk is the wavelength For these samples we will have
17
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
to locate all critical points and establish how they are connected by the flow linesThis will require very high precision computations since the flow lines can passvery close to the other saddle points The result of this stage will be the samplingof the random graph that was described above
bull In this stage we are going to study the percolation on the random graph thatwas sampled in the first stage We will have to sample sufficient number of thepercolation realizations on each of the graph samples and study their statisticsFinally it will be compared with the known statistics for the nodal domains ofRPW
213 Numerical Solution of Equations in Biochemistry
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells can be described in terms of partial differential equa-tions (PDEs) and master equations for probability distributions of biochemical speciesinvolved These equations include the chemical master equation (CME) and the chem-ical Fokker-Planck equation (CFP) which are introduced in Special Topic Course [1]Numerical solution of these equations is challenging due to the structure and size ofparticular problems
The CME is an infinite system of linear differential equations which has to be truncatedto a finite size for computational reasons The CFP equation is a linear evolutionaryPDE of convection-diffusion type which can be solved by standard numerical methodssuch as the finite difference method (FDM) finite volume method (FVM) and finiteelement method (FEM) [23] In any case approximate stationary solution of boththe CME and CFP is determined by a null-space of a large sparse and nonsymmetricmatrix Computing a null-space is not straightforward however it can be solved bystandard methods of numerical linear algebra [345]
18
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
In this project we will first investigate the properties of CME and CFP equations andthen concentrate on their efficient numerical solution This is particularly challengingin the case of a higher number of chemical species (more than three) in the systembecause this number corresponds to the dimension of the resulting problem Solvinghigh-dimensional problems is difficult due to the so called ldquocurse of dimensionalityrdquomeaning that the number of degrees of freedom grows exponentially with the dimensionHowever this can be countered by using tensor methods [6]
This project is suitable for students interested in numerical methods for partial differ-ential equations The focus will be on numerical methods rather than on applicationsin biology
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] H C Elman D J Silvester A J Wathen ldquoFinite Elements and Fast IterativeSolvers With Applications in Incompressible Fluid Dynamicsrdquo 2005
[3] Core Course B2 ldquoFinite Element Methods and Further Numerical Linear Algebrardquo
[4] L N Trefethen D Bau ldquoNumerical Linear Algebrardquo 1997
[5] Core Course B1 ldquoNumerical Solution of Differential Equations and Numerical LinearAlgebrardquo
[6] V Kazeev M Khammash M Nip and C Schwab ldquoDirect Solution of the ChemicalMaster Equation using Quantized Tensor Trainsrdquo 2013
214 Numerical Solution of the Rotating Disc Electrode Problem
Supervisor Dr Kathryn GillowContact gillowkmathsoxacuk
The basic idea of an electrochemical experiment is that a known potential is appliedto a working electrode in a solution This causes oxidation or reduction to take placeat the electrode and in turn this means that a current (which can be measured) flowsThe current depends on a number of physical parameters of the solution including thediffusion coefficient the resistance of the solution and the rate of reaction
Mathematically the concentration of the chemicals is modelled using a reaction-convection-diffusion equation and for a rotating disc electrode we can assume that one space dimen-sion is enough for the model The current is then a linear functional of the concentrationSolving for the current with given values of the parameters is known as the forwards prob-lem Of more interest is the inverse problem where an experimental current is given andand the parameters are to be calculated
The idea of this project is to first develop an efficient solver for the forwards problemand then use it as the basis of a solver for the inverse problem It is then of interest tofind out how the experimentally controllable parameters (reaction rate applied potential
19
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
etc) affect the performance of the inverse problem solver
20
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
3 Biological and Medical Application Projects
31 Circadian Rhythms and their Robustness to Noise
Supervisors Dr Tomas Vejchodsky and Dr Radek ErbanContact vejchodskymathsoxacuk and erbanmathsoxacuk
Biochemical processes in living cells typically involve chemical species of very low copynumbers (eg one molecule of DNA and low numbers of mRNA molecules) Thereforethe classical description based on concentrations is not applicable The intrinsic noiseis crucial in these systems because it yields substantial and important effects such asstochastic focusing stochastic resonance and noise-induced oscillations [1]
In this project we will try to explain how regular circadian rhythms can robustly persistin biochemical systems that are highly influenced by the intrinsic noise There are manymathematical models of circadian rhythms based on gene regulation [234] This meansthat concentrations of certain proteins within the cell cytoplasm oscillates within a 24hour period due to positive andor negative feedback loops The feedback is caused bythe protein molecule binding to the promoter region of a gene which activates or repressesthe gene expression The intrinsic noise can have strong effects on these biochemicalreactions because of low copy numbers of interacting biomolecules involved In spite ofthis fact robust circadian rhythms are observed in many types of cells
In this project we begin with model reduction of a mathematical model of circadianrhythms using the quasi-steady state assumptions [5] Then the reduced system willbe analysed for bifurcations to understand details of its dynamics and sensitivity tonoise We will use numerical methods to solve systems of ordinary differential equationsand stochastic simulation algorithms to sample trajectories of stochastic systems Wewill aim to explain details of the circadian model dynamics in both deterministic andstochastic regimes This will yield the understanding of the observed noise robustness
21
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
of circadian rhythms
References
[1] Special Topic Course C64b ldquoStochastic Modelling of Biological Processesrdquo
[2] D B Forger and C S Peskin A detailed predictive model of the mammalian circa-dian clock PNAS 100(25) 14806-14811 2003
[3] J Villar H Kueh N Barkai S Leibler Mechanisms of noise-resistance in geneticoscillators Proc Nat Acad Sci USA 99 pp 5988-5992 2002
[4] Z Xie D Kulasiri Modelling of circadian rhythms in Drosophila incorporating theinterlocked PERTIM and VRIPDP1 feedback loops J Theor Biology 245 290-3042007
[5] L A Segel and M Slemrod The quasi-steady-state assumption a case study inperturbation SIAM Review 31(3) 446-477 1989
32 The Analysis of Low Dimensional Plankton Models
Supervisor Dr Irene MorozContact morozmathsoxacuk
The increasing exploitation of marine resources has driven a demand for complex bio-geochemical models of the oceans and the life they contain The current models areconstructed from the bottom up considering the biochemistry of individual species orfunctional types allowing them to interact according to their position in the food weband embedding the ecological system in a physical model of ocean dynamics The re-sulting ecology simulation models typically have no conservation laws and the ecologyoften produces emergent properties that is surprising behaviours for which there isno obvious explanation Because realistic models have too many experimentally poorlydefined parameters (often in excess of 100) there is a need to analyse simpler models
A recent approach by Cropp and Norbury (2007) involves the construction of complexecosystem models by imposing conservation of mass with explicit resource limitation atall trophic levels (ie positions occupied in a food chain) The project aims to analysemodels containing two ldquopredatorsrdquo and two ldquopreyrdquo with Michaelis-Menten kinematicsA systematic approach to elicit the bifurcation structure and routes to chaos usingparameter values appropriate to different ocean areas would be adopted In particularthe influence of nonlinearity in the functional (life) forms on the stability properties of thesystem and the bifurcation properties of the model will be comprehensively numericallyenumerated and mathematically analysed
22
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
33 Individual and Population-Level Models for Cell Biology Processes
Supervisor Dr Ruth BakerCollaborator Dr Mat Simpson Queensland University of Technology Bris-baneContact bakermathsoxacuk
Modelling the individual and collective behaviour of cells is central to many areas oftheoretical biology from the development of embryos to the growth and invasion oftumours Methods for modelling cell processes at the individual level include agent-based space-jump and velocity-jump processes both on- and off-lattice One may includebiological detail in these models taking volume exclusion into account by for exampleallowing a maximum occupancy of lattice sites or modelling adhesion by allowing cellmovement rates to depend on the local cell density However it is often difficult tocarry out mathematical analysis of such models and we are restricted to computationalsimulation to generate statistics on population-level behaviour Models derived on thepopulation level whilst more amenable to analysis are often more phenomenologicalwithout careful regard given to the detail of the cell processes under consideration Therigorous development of connections between individual- and population-level models iscrucial if we are to accurately interrogate biological systems
A project in this area could investigate a number of phenomena in relation to theseprocesses not limited to the following
bull The links between exclusion processes (where a lattice site may be occupied by atmost one agent) and those that allow multiple agents to occupy the same site
bull The extent to which limiting PDEs describing the evolution of cell density can bederived from different underlying motility models
bull The effects of cell shape on motility and proliferation
bull The effects of crowding upon cell processes and the possibility for anomalousdiffusion
bull The potential of exclusion processes to give rise to patterning by a Turing-typemechanism
bull The extension of velocity-jump and off-lattice models to include domain growthand comparison of results with those already put forward in the literature
References
[1] M J Simpson R E Baker and S W McCue Models of collective cell spreadingwith variable cell aspect ratio A motivation for degenerate diffusion models Phys RevE 83(2) 021901 (2011)
[2] R E Baker C A Yates and R Erban From microscopic to mesoscopic descriptionsof cell migration on growing domains Bull Math Biol 72(3)719-762 (2010)
23
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
x100 300200
HORIZONTAL POPULATION
0
1
ChltChgt
x100 300200
VERTICAL POPULATION
0
1
CvltCvgt
t= 500
x100 300200
y1
20
t = 0
x100 300200
y1
20
(d)
(b)
(c)
(e)
(a)
(ij)
(ij+1)
(ij-1)
(i-1j)
(i-1j+1)
(i-1j-1)
(i+1j)
(i+1j+1)
(i+1j-1)
(i+2j)
(i+2j+1)
(i+2j-1)
x
y
Figure 8 Comparing the motility of agents of aspect ratio L = 2 undergoing a randomwalk with rotations
[3] M J Simpson K A Landman and B D Hughes Multi-species simple exclusionprocesses Physica A 388(4)399-406 (2009)
34 A New Model for the Establishment of Morphogen Gradients
Supervisor Dr Ruth BakerCollaborator Professor Stas Shvartsman University of PrincetonContact bakermathsoxacuk
During embryonic development a single cell gives rise to the whole organism wherecells of multiple different types are arranged in complex structures of functional tissuesand organs This remarkable transformation relies on extensive cell-cell communicationIn one type of cell communication a small group of cells produce a chemical that in-structs cells located nearby Cells located close to the source of the signal receive a lotof it whereas cells located further away receive progressively smaller amounts In thisway a locally produced chemical establishes a concentration profile that can ldquoorganizerdquothe developing tissue providing spatial control of gene expression and cell differentia-tion Starting from the late 1980s such concentration profiles known as ldquomorphogengradientsrdquo have been detected in a large number of developing tissues in essentially allanimals from worms to humans
24
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
One of the most popular models for gradient formation is based on the localized produc-tion and spatially uniform degradation of a diffusible protein In this model moleculesmove to the cells which are ldquowaitingrdquo for the arrival of a signal that tells them whatto do Mathematically the system is often modelled using reaction-diffusion equationswhich can be readily solved and used to fit to experimental data Recently howeverexperiments in a number of systems suggest that the mechanisms underlying morphogengradient formation can be more complex Instead of passively waiting for the arrival ofthe signal cells can form long-range dynamic projections that reach out in space andare used to transport a signal back to the cell The resulting concentration profile is thesame but the mechanism of formation is very different
This project is concerned with formulating a theory of morphogen gradient formationby these dynamic projections known as ldquocytonemesrdquo The start point will be a one-dimensional model of cytoneme-mediated chemical transport based on the theory of twointeracting random walks which describe both moving cytonemes and moving moleculesFurther extensions will include the extension to two spatial dimensions and the incorpo-ration of further important details from cell biology These models will be analyzed us-ing a range of computational and analytical tools from stochastic simulations to Greensfunctions techniques
In parallel with this theoretical work the Shvartsman Lab are investigating the existenceand potential roles of cytonemes in cell communication mediated by the EpidermalGrowth Factor signaling pathway which controls developmental processes in multipleanimals Our experimental system is Drosophila where the EGF pathway is involved inpatterning of essentially all tissue types
Figure 9 Cells from the anterior region of a Drosophila wing disc projecting cytonemesin an in vitro experiment
25
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
35 Modelling the Regrowth and Homoeostasis of Skin
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statement There is considerable interest in understand-ing how human skin reforms after damage and how its thickness is controlled Suchunderstanding is necessary for example for improving methods of skin grafts for burnsvictims for identifying methods of controlling skin diseases such as psoriasis and for as-sisting in the creation of artificial skin to test the safety of household products cosmeticsand new drugs Substantial experimental evidence comes from groups in Brisbane andUtrecht who are examining the behaviour and viability of artificial skin They remove allthe cells from skin samples and then deposit a few cells in the tissue before incubating itin a well-defined medium Examples of the type of regrowth that they observe are shownin the diagram below Three distinct layers can be detected the lower de-epithelialisedhuman dermis (DED) layer which is the extracellular material from the original samplewith no cells in it the viable epidermal layer (TAL) in which the deposited cells dividemove and grow and the cornified layer (KL) which contains dead cells that still retainsome structure Interesting behaviour can be observed for example the DED region hasan undulating surface yet the KL layer is very flat The aim of this project is to under-stand the dynamics of the growing layers and the mechanisms that might be controllingthe observed behaviour There is considerable discussion and controversy about how thecells communicate and how the layers are formed the models developed in this projectwill be used to test and compare the alternative hypotheses
Description of the planned approach and the techniques needed Models will beexamined and extended which involve transport of chemicals and cell motion and requirethe introduction of moving boundaries to account for the various interfaces separating thedifferent skin layers and their changing thickness Mechanisms to be considered couldinclude growth motion and death of cells transport of nutrient and other signallingmolecules mechanical stresses in the layers The behaviour of the resulting models willbe examined using a combination of analytical and numerical approaches The projectwill involve close collaboration with Dr Jos Malda University of Utrecht
References
[1] R A Dawson Z Upton J Malda and D G Harkin (2006) Transplantation 81(12)1668-1676
26
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
[2] G Topping J Malda R A Dawson and Z Upton (2006) Primary Intention 1414-21
[3] M Ponec (2002) Advanced Drug Delivery Reviews 54 S19-S30
[4] H J Stark K Boehnke N Mirancea et al (2006) Jl Invest Derm Symp Proc 11(1)93-105
36 Modelling the Growth of Tumour Spheroids
Supervisors Prof Helen Byrne and Prof Colin PleaseContact byrnemathsoxacuk and pleasemathsoxacuk
Background and problem statementA critical step in the dissemination of ovarian cancer is the formation of multicellularspheroids from cells shed from the primary tumour There is increasing evidence thatthe mechanical properties of the tissue surrounding such tumour spheroids may influencetheir ability to grow and spread In this project the student will develop mathematicalmodels describing the growth of multicellular spheroids in established bioengineeredthree-dimensional (3D) microenvironments for culturing ovarian cancer cells in vitroThe project will be supported by experimental work being conducted at the QueenslandUniversity of Technology Brisbane Australia The data obtained (see figure below)demonstrates that cells cultured in gels form spherical clusters of different sizes andthat their size depends on the mechanical properties of the tissue in which the cells arelocated
The aim of this project is to develop and analyse new continuum models that can be usedto investigate how cell-cell and cell-tissue interactions and the mechanical properties ofthe gel in which the spheroid is embedded influence tumour invasion The project couldhave a mathematical or numerical focus (or involve a combination of the two)
References
[1] C Gang J Tse RK Jain and LL Munn (2009) PLoS ONE 4(2) e4632
[2] C Y Chen H M Byrne and J R King (2001) J Math Biol 43 191-220
[3] S Krause M V Maffini A M Soto and C Sonnenschein (2008) Tissue Eng PartC Methods 14(3)261-71
27
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
37 DiscreteHybrid Modelling of Lymphangiogenesis
Supervisors Prof Helen Byrne and Dr Chris BellContact byrnemathsoxacuk and bellmathsoxacuk
Background and problem statement The lymphatic and vascular systems are cou-pled fluid and nutrients are delivered by the vasculature while extracellular fluid flowsfrom the capillaries into the lymphatic microvessels and is returned to the vasculaturesystem via the thoracic ducts Failure of the lymphatic system can result in conditionssuch as lymphoedema
Although both transport systems interact and are similar comprising large networks ofvessels with an endothelial lining experimental and theoretical research has focussed onthe blood system A variety of theoretical frameworks have been used to study aspectsof angiogenesis and vasculogenesis (the de novo formation of new blood vessels) [Perfahlet al 2011] Modelling of the lymphatic system is less advanced Roose amp Fowler (2008)considered the pre-patterning of lymphatic vessel morphology within collagen cells viathe establishment of a fluid flow network while Friedman amp Lolas (2005) considered areaction-diffusion equation for lymphangiogenesis which neglects biomechanical stimuli
Recently Swartz and coworkers have developed novel assays for the detailed investiga-tion of network formation from blood endothelial cells (BECs) and lymph endothelialcells (LECs) Ng Helm amp Swartz (2004) exposed ECs to interstitial flow in collagen gelsand found key differences between the two cell types in their cell-cell and cell-matrix in-teractions and their responses to the local biophysical environment Through combinedexperimental and theoretical work Helm et al (2005) and Fleury et al (2006) showedthat interstitial flow affects LEC and BEC organization in a fibrin matrix with matrix-bound vascular endothelial growth factor (VEGF) Helm Zisch amp Swartz (2007) foundthat extracellular matrix composition (fibrin versus collagen) differentially influences theorganization of the two endothelial cell types with LECs showing the most extensiveorganization in fibrin-only matrix and BECs preferring a collagen matrix These differ-ences are also observed in vivo and it is hypothesised that during dermal wound healingthe tissue matrix remodels so that initially it is optimised for angiogenesis and at laterstages for lymphangiogenesis
The aim of this project is to generate a predictive tool that can be used to inform networkformation from lymph endothelial cells in vivo (with applications to wound healing) andin vitro (with applications to tissue engineering for example)
Description of the Planned Approach and the Techniques Needed In thisproject we will use a discretehybrid modelling approach similar to that developedin (Owen et al 2009) and (Perfahl et al 2011) to study lymphangiogenesis and theinterplay between the lymph and vascular networks In more detail a discrete modelthat accounts for the evolving spatial structure of the vascular network will be coupledto reaction-diffusion equations describing the distribution of key growth factors Thebehaviour of the model will be investigated using a combination of analytical and nu-merical approaches The models will be informed by the experimental results of Swartzand co-workers and once formulated will be validated against the experimental data
28
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
Figure 10 3D lymphangiogenesis assay Cells sprout from dextran beads embedded infibrin gel
References
[1] Perfahl H Byrne HM Chen T Estrella V Alarcon T et al (2011) PLoS ONE 6(4)e14790
[2] Roose T Fowler AC (2008) Bulletin of Mathematical Biology 70 1772ndash1789
[3] Friedman A Lolas G (2005) Math Mod Meth Appl Sci 15 95107
[4] Ng CP Helm C-LE Swartz MA (2004) Microvas Res 68 258ndash264
[5] Helm CL Fleury ME Zisch AH Boschetti F Swartz MA (2005) Proc Natl AcadSci USA 102 15779ndash15784
[6] Fleury ME Boardman KC Swartz MA (2006) Biophys J 91 113ndash121
[7] Helm CL Zisch A Swartz MA (2007) Biotechnol Bioeng 96 167ndash176
[8] Owen MR Alarcon T Maini PK Byrne HM (2009) J Math Biol 58 689ndash721
38 Mathematical Modelling of the Negative Selection of T Cells inthe Thymus
Supervisor Prof Jon ChapmanContact chapmanmathsoxacuk
Background The thymus is the primary organ for the generation of naive T cellsDuring their maturation T cells acquire an antigen-receptor with a randomly chosenspecificity including reactivity to the bodyrsquos own proteins To purge this pool of im-mature T cells from cells with a reactivity to self-antigens specialised epithelial cellsin the medulla of the thymus produce a broad range of proteins which are normallyonly detected in differentiated organs residing elsewhere in the body The efficient andgenome-wide transcription of these so called self-antigens secures the completeness by
29
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
which these self-antigen reactive T cells are deleted Hence thymic medullary epithelialcells as a population provide a comprehensive ldquomolecular libraryrdquo of self-antigens thatwhen recognized by developing self-reactive T cells will initiate their death This dele-tion of potentially harmful T cells is known as thymic negative selection and preventsthe formation of a repertoire of effector T cells able to initiate an injurious autoimmuneresponse
The range of promiscuous genes expressed by each single medullary thymic epithelial cell(mTEC) is however thought to be limited to a selection of self-antigens Consequentlythe library of self-antigens would only be representative in its entirety when a largernumber of these medullary epithelial cells are concurrently available However a detailedquantitative and qualitative analysis of this concept has not yet been accomplished
Description of the planned approach and the techniques needed This projectis to investigate mathematical models of T cell negative selection Some existing modelsconsider just one T cell-mTEC interaction but include multiple receptors with somethreshold criteria for whether the interaction ldquofiresrdquo [1] Other models consider just onereceptor-ligand binding but in more detail incorporating the sequence of the receptorpeptide into the model so that the strength of the interaction is determined by thesimilarity between the receptor sequence and the ligand sequence [2] The goal of theproject is to synthesis key components of existing models in such a way that they aresuitable for the incorporation of gene expression data from individual mTECs
The mathematics will involve stochastic models of reactionsinteractionsbinding-unbindingAn understanding of elementary probability theory and ordinary differential equationswill help
Reasonable expected outcome of project A new model for T cell negative selec-tion
References
[1] Berg HA van den Rand DA Burroughs NJ ldquoA reliable and safe t cell repertoirebased on low-affinity t cell receptorsrdquo Journal of Theoretical Biology 209465ndash486 2001
[2] Detours Vincent Mehr Ramit Perelson Alan S ldquoA quantitative theory of affinity-driven t cell repertoire selectionrdquo Journal of Theoretical Biology 200389ndash403 1999
39 The Dynamics and Mechanics of The Eukaryotic Axoneme
Supervisors Dr Eamonn Gaffney and Dr Hermes GadelhaContact gaffneymathsoxacuk and gadelhamathsoxacuk
Background The eukaryotic axoneme is a ubiquitous organelle found within cilia andflagella which are filamentous cell appendages whose beating drives fluids in numerousphysiologically important settings including sperm swimming and egg transport in re-production mucociliary clearance within the lung circulation within the cerebrospinalfluid system symmetry breaking in early developmental biology and the virulence of
30
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
numerous medically important pathogenic parasites Dynein molecular motors contractwithin the axoneme exerting internal forces and moments mechanically these are bal-anced by a combination of viscous drag from the medium surrounding the cell and apassive elastic restoring response of the cilium or flagellum The resulting dynamicsgives rise to a propagating waveform which drives the surrounding fluid and for freecells results in swimming Nonetheless the subcellular details of the collective behav-ior of the dyneins and their regulation are poorly understood suffering from numerouscompeting hypotheses However consider the combination of cell videomicroscopy andmechanics From movies of a swimming cell for example one can determine rate of vis-cous dissipation associated with the flagellum using fluid dynamical theory By energyconservation this is the time averaged rate of working of the dyneins allowing energyexpenditure to be measured as one example of how mechanics can be extracted frommicroscopy and how ultimately our understanding of the mechanics and biology drivingcellular swimming may be improved
Reasonable expected outcome of project There are many possible projects andthus outcomes One example would be to investigate image analysis techniques to im-prove flagellar extraction another to explore sperm filament mechanics in detail usingmicromanipulator experiment data a third would involve the use of fluid and filamentmechanics to assess dynein behaviour from current videomicroscopy data and a finalexample would be to explore which waveforms are the most energetically efficient
Techniques Depending on the detailed choice of project investigations in this fieldcould rely on calculus of variations and the numerical solution of partial differentialequations for novel image analysis Alternatively the project could focus on viscousfluid mechanics and elastic micromechanics to assess dynein behaviours from microscopevideos or combine mechanics and calculus of variations to find optimal waveforms forswimming
References
[1] H Gadelha E A Gaffney and A Goriely PNAS 3012180-85 2013
[2] E A Gaffney H Gadelha et al Ann Rev Fluid Mech 43501-28 2011
31
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
4 Physical Application Projects
41 Swarm Robotics From Experiments to Mathematical Models
Supervisors Dr Ulrich Dobramysl and Dr Radek ErbanContact dobramyslmathsoxacuk and erbanmathsoxacuk
Much theory has been developed for the coordination and control of distributed au-tonomous agents where collections of robots are acting in environments in which onlyshort-range communication is possible [1] By performing actions based on the presenceor absence of signals algorithms have been created to achieve some greater group leveltask for instance to reconnoitre an area of interest whilst collecting data or maintain-ing formations [2] Algorithms of swarm (collective) robotics have often been motivatedby collective animal behaviour [3] Collective animal behaviour has been of interest formathematical research throughout the last century [4] In many of these mathematicalapproaches a model is proposed and then compared to the real-world behaviour of theanimal groups under certain comparison measures One example of this type of modelwas developed by Couzin et al [5] for individuals communicating through visual andcontact interactions Depending on parameter values it can generate directed swarmstorus movement or weakly ordered groups of animals
In the Mathematical Institute we have a group of mobile e-puck robots [6] which can in-teract through a number of different channels (audio video bluetooth) mdash see Figure 11These robots have sensors (resp actuators) for contact and visual communication whichcan be used for mimicking the behaviour of animal models in [45] In a previous dis-sertation [7] an MSc student investigated an implementation of searching algorithmssimilar to those used by flagellated bacteria in a robotic system A paper based on thisMSc dissertation [7] is currently being prepared for publication
In this project we will use a combination of experiments with robots and mathematicalmodelling We will investigate accurate and efficient ways to mathematically model col-lective behaviour of individuals (robots) communicating through short-range (proximitysensors) and long-range (auditory and visual cues bluetooth) means We would liketo understand the advantages of different types of hierarchies and strategies within agroup of robots for the successful completion of a pre-defined group task similar to theresearch that has been done in [8] for hens inside a barn and in [9] for pigeons during aflight Depending on student interest the robot tasks can also involve target area find-ing maintaining formations in a complicated geometry or other assignments [10] Thisproject will involve analytical and numerical modeling microcontroller programmingand efficient sensor data analysis
References
[1] J Reif and H Wang Robotics and Autonomous Systems 27(3)171194 1999
[2] J Desai J Ostrowski and V Kuma IEEE Transactions on robotics of automation17(6)905908 2001
32
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
Figure 11 A group of mobile e-puck robots
[3] Garnier S in Bio-inspired self-organizing robotic syst eds Meng and Jin Springerpp 105-120 (2011)
[4] Sumpter D Collective Animal Behavior Princeton University Press (2011)
[5] Couzin I et al Journal of Theoretical Biology 218 pp 1-11 (2002)
[6] Mondada F et al Proc of 9th Conf on Autonomous Robot Systems and Competi-tions 1 pp 59-65 (2009)
[7] JTKing ldquoHard-Sphere Velocity-Jump Processes Applications to Swarm RoboticsrdquoMSc dissertation 2013
[8] Linquist B Bulletin of Mathematical Biology 71 pp 556-584 (2009)
[9] Nagy M et al Nature 464 pp 890-893 (2010)
[10] Gazi V and Passimo K Swarm Stability and Optimization Springer (2011)
42 A Simple Model for Dansgaard-Oeschger Events
Supervisors Dr Ian Hewitt and Dr Andrew FowlerContact hewittmathsoxacuk and fowlermathsoxacuk
Many northern hemisphere climate records show a series of rapid climate changes thatrecurred throughout the last glacial period These ldquoDansgaard-Oeschgerrdquo (D-O) se-quences are most prominent in Greenland ice cores and consist of a very rapid (decades)warming followed by an initial slow cooling and a final rapid temperature fall Theyoccurred somewhat periodically with a period of around 1500 years What is respon-sible for this sequence is of course of great interest given current climate changes and
33
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
it has been hotly debated Various factors point towards changes in ocean circulationbeing key The suggestion is that the Atlantic meridional overturning circulation mdash theconveyor that transports warm equatorial water northwards and keeps the UK warm mdashunderwent sudden changes in strength and that this caused the rapid changes in airtemperature over the Northern Hemisphere A sudden injection of fresh water into theNorth Atlantic may be sufficient to cause such switches but it is not fully understoodwhat would cause this
This project would explore the hypothesis that the Dansgaard-Oeschger events occuras a self-sustained oscillation of the ocean dynamics and the Northern hemisphere icesheets In this mechanism the melting and growth of the ice sheets would be determinedby the strength of the ocean circulation but at the same time the melting itself providesthe fresh water that drives the ocean circulation
The project would consist of constructing simplified box models of the ocean and theice sheets These can be reduced to systems of non-linear ordinary differential equationsthat can be solved numerically and analyzed to examine steady states stability limitcycles etc under different assumptions There are numerous levels of complexity mdashboth in the modelling and mathematical analysis mdash that can be added sequentiallydepending on time and earlier success
43 Modelling Snow and Ice Melt
Supervisor Dr Ian HewittContact hewittmathsoxacuk
On ice sheets and glaciers snow builds up on the surface over the winter and meltsduring the summer The quantity of the melt water that runs off from the surface isimportant because it is a large component of sea-level rise Predicting this run off is notas straightforward as might be imagined because as the snow melts the water infiltratesinto the snowpack and some of it refreezes while the rest runs off along the ice surfaceAt lower altitudes there is more melting than snow so the snowpack is exhausted bymid summer and the underlying ice also melts Almost all of the water runs off in thiscase At high altitudes the amount of melting is less than the accumulation of snow soeach year the older layers of snow are gradually compacted to form ice Here it is verypoorly understood how much of the water runs off and how much refreezes
The aim of this project is to derive a mathematical model for the melting and compactingsnow pack and to solve some simplified problems to understand the roles of differentphysical parameters in influencing the amount of run off The first task will be to developa model It will be a continuum model describing snow as a deformable porous mediumwith Darcy flow through the pores The interesting and unusual aspect of the model isthe refreezing which will require incorporating an energy conservation equation Themodel will be simplified and then solved in one dimension (vertical) for some simpleboundary conditions This will almost certainly require a combination of numerical andasymptotic methods
This project would require an interest in continuum modelling and fluid mechanics
34
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
a willingness to engage with and translate physics into mathematics and some open-mindedness about using different techniques for solving partial differential equations
There are various other potential projects associated with ice sheets and modelling in-teresting processes that affect them Please discuss with me
44 A Network-Based Computational Approach to Erosion Modelling
Supervisor Dr Ian HewittContact hewittmathsoxacuk
Many erosive processes produce interesting geometrical structures and a common resultis the formation of branching channel networks The obvious example is river networksbut similar processes occur in erosion of limestone caves groundwater flow through soilporous flow in oil and gas reservoirs and the flow of molten rock inside the Earth In allthese situations fluid flow over or through a porous substrate causes erosion that feedsback to alter fluid flow
One of the interesting challenges of modelling this process mathematically is that thefluid flow transitions from an initially uniform state to an evolved state with a vastlydifferent structure For instance a porous rock in which fluid flow is modelled by Darcyrsquoslaw may be eroded to form a cylindrical conduit for which Darcyrsquos law is no longerappropriate
This project will explore a new numerical method to describe these processes by com-bining a distributed porous domain with a network of localized one-dimensional flowelements The simplest generic problem involves an elliptic partial differential equationto describe fluid conservation coupled to an evolution equation (ODE) for the erodi-ble material The method to be developed will use a finite element method to solvethe equations incorporating elements of different dimension to account for the differentflow structures A similar approach has yielded realistic ldquolookingrdquo results for water flowbeneath a glacier [1] but has raised a number of interesting questions that need to beexplored
The project will involve getting to grips with physical principles of fluid flow and erosiondeveloping and coding a numerical model and exploring its behaviour
Reference
[1] M A Werder I J Hewitt C Schoof and G E Flowers Modeling chanelized anddistributed subglacial drainage in two dimensions J Geophys Res 118 1ndash19 2013
45 Retracting Rims
Supervisor Dr Andreas MunchContact muenchmathsoxacuk
These are actually three projects which share some common features
35
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
I Bursting films When a freely suspended film of a viscous liquid ruptures (forexample in a bursting bubble) a rim forms around the expanding hole that grows as itis pushed deeper into the yet unperturbed film Moreover the rim forms ondulations inthe spanwise direction This problem has a long history with the original Taylor-Culickformula describing the retraction velocity based on conservation of mass and momentum(However a systematic derivation of the long time evolution of the film profile has notbeen carried out for the case of large viscous dissipation) The task in the thesis willbe to (a) Rederive the underlying thin film model (b) investigate the evolution of thecross section using a (self-written) matlab code and asymptotic analysis (c) investigatethe stability of the rim using a linear stability analysis Extensions from the planar tothe axisymmetric case may also be considered and the effect of viscoelasticity
II Dewetting rims with strong slip When a liquid is repelled from a flat surfaceie it is hydrophobic holes will grow once they are formed since this reduced the totalenergy of the liquid films (ie the sum of all interface energies is reduced by collectingthe liquid into ridges or droplets with only a small liquidsolid interface area) Thedynamics of this process has been carefully investigated for thin polymeric liquids andit can show a surprisingly rich behaviour in particular in the case where there is alsosignificant effective slip at the interface In this project we will (a) redrive the underlyingthin film model (b) investigate the evolution of the cross section using a (self-written)matlab code and asymptotic analysis Further steps could involve a stability analysisandor the inclusion of visco-elasticity
III Inertial dewetting When a low-viscosity liquid such as water is deposited asa film of thicknesses around 1mm onto a very hydrophobic surface such as Teflon theopening of a hole leads to a very fast retraction of the liquid as it dewets from thesubstrate The Reynolds numbers are large than one suggesting inertia is important(more as in I for low-viscosity films than as in II) In contrast to the bursting suspendedfilms the normal component of gravity and friction at the liquidsolid substrate enterFocusing first on the effect of the former we will (a) derive a model for this situation(b) investigate the wave structure to identify the two fronts that are observed in theexperiments (c) compare with the experiments The step (b) will involve writing amatlab code to solve the model equations
Extensions could be going from the planar- to the axisymmetric situation or includingthe effect of friction at the substrate
46 Modelling Spray Deposition for Applications in Manufacturing Su-percapacitors
Supervisors Dr Andreas Munch and Dr Jim OliverContact muenchmathsoxacuk and olivermathsoxacuk
One of the main techniques to design nano-structured porous electrodes for use in super-capacitors and batteries is spray coating of colloidal droplets When a colloidal droplet
36
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
impacts a heated solid surface it will spread to a maximal extent while the liquid evap-orates leaving behind nano-patterns of colloidal particles A basic continuum model forthe spreading and evaporating colloidal droplet may be found in [1] The model involvesa system of partial differential equations coupling the fluid motion with the concentra-tion field of the particle distribution Combining asymptotic and numerical techniquesthe project aims to derive simplified models for the droplet profile and the evolutionof the volume fraction that are used to predict the particle distribution under variousimpact conditions (The project may involve collaboration with the research group ofProfessor Patrick Grant in Oxfordrsquos Materials Sciences Department)
Reference
[1] K L Maki et al Langmuir 27(18)11347-11363 2011
47 Mathematical Modelling of Membrane Fouling for Water Filtra-tion
Supervisors Dr Ian Griffiths and Dr Andreas MunchContact iangriffithsmathsoxacuk and muenchmathsoxacuk
Understanding membrane fouling is a key goal in separation science and is an area inwhich detailed mathematical modelling can provide key insight for membrane designoptimization Historically in a typical filtration set-up there are four key membranefouling mechanisms
bull Standard blocking mdash small particles pass into the membrane pores and a finitenumber adhere to the walls causing pore constriction
bull Partial blocking mdash larger particles land on the membrane surface and partiallycover a pore
bull Complete blocking mdash larger particles land on the membrane surface and cover apore entirely
bull Caking mdash a layer of particles builds up on the membrane surface following completeblocking which provides a further resistance in the form of an additional porousmedium through which the feed must also permeate
Recently new asymmetric membranes have been developed whose pore radius varies withdepth Such membranes have been demonstrated to possess novel filtration propertiesFor instance a membrane whose pores constrict with depth can capture different particlesizes at different positions while a membrane whose pores expand with depth mayoffer a mechanism to control the surface build-up associated with caking This projectaims to understand the role of each of the fouling mechanisms in the clogging of anasymmetric membrane and in particular the interplay between the various mechanismsMathematical models based on stochastic simulations will be developed and continuumdescriptions will be derived by examining various limiting time-averaged cases The
37
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
outcome will be a series of mathematical models that make predictions on the optimaloperating regimes to filter a given contaminant
References
[1] G R Bolton D LaCasse and R Kuriyel 2006 Combined models of membrane foulingDevelopment and application to microfiltration and ultrafiltration of biological fluids JMemb Sci 277 75ndash84
[2] G R Bolton A W Boesch and M J Lazzara 2006 The effects of flow rate on mem-brane capacity Development and application of adsorptive membrane fouling modelsJ Memb Sci 279 625ndash634
[3] C-C Ho and A L Zydney 2000 A combined pore blockage and cake filtration modelfor protein fouling during microfiltration J Colloid Interf Sci 232 389ndash399
48 Flow-Induced ldquoSnap-Throughrdquo
Supervisors Dr Dominic Vella and Dr Derek MoultonContact vellamathsoxacuk and moultonmathsoxacuk
It is well known that in high-speed winds umbrellas are forced from their initial stateto an inverted state that is less efficient at keeping the rain off This ldquosnap-throughrdquoinstability is an intrinsic feature of elastic systems with a natural curvature and is usedin biology and engineering to generate fast motions The proposed project looks atthe combination of elasticity and fluid flows that produces this instability and aims tounderstand the critical properties of the transition
The project will begin by reviewing previous work on the ldquosnap-throughrdquo instability ofan elastic arch subjected to static loads [1-3] It will then move on to understand underwhich conditions fluid loading (eg in the limits of high and low speed flows) can cause
38
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
such a snap-through to occur the dynamics of this snapping and whether the snapped-through state is stable We hope to include comparisons between models developed aspart of this project and experiments conducted at Virginia Tech
Expected outcomes involve the identification of a dimensionless fluid loading parameterand quantifying when snapping should occur as a function of this parameter
References
[1] J S Humphreys On dynamic snap buckling of shallow arches AIAA J 4 878 (1967)
[2] A Fargette S Neukirch and A Antkowiak Elastocapillary Snappinghttparxivorgabs13071775
[3] A Pandey D E Moulton D Vella and D P Holmes Dynamics of Snapping Beamsand Jumping Poppers httparxivorgabs13103703
49 Plumes with Buoyancy Reversal
Supervisors Dr Dominic Vella and Prof John WettlauferContact vellamathsoxacuk and wettlaufermathsoxacuk
One means by which the worst effects of climate change might be avoided is to pumplarge amounts of carbon dioxide into sub-surface aquifers so-called carbon sequestration[1] When pumped into aquifers the carbon dioxide remains buoyant with respect to theambient liquid and so rises back towards the surface In practice this rise is halted bylayers of relatively impermeable rock which trap the carbon dioxide until it has sufficienttime to dissolve in the ambient water However once dissolved the carbon dioxidewatermixture is unusual because it becomes denser than the water it will therefore reversedirection and sink
In an unconfined porous medium it might be expected that the original plume mayactually reach a steady height sufficient mixing should occur over the course of its risethat the source of buoyancy becomes extinct at some critical height This project willaddress the question does an unconfined plume have a maximum rise height
The project will begin by reviewing the classic analyses of buoyant plumes in a porousmedium [2] and developing a numerical code to verify the similarity solutions presentedthere Using a more realistic equation of state for carbon dioxidewater mixtures (in-corporating buoyancy reversal) in the previously developed code would then allow thestudent to answer the question of whether a maximum rise height exists Depending ontime the final component of the project would be an experimental realisation of thisphenomenon in the Mathematical Observatory
References
[1] M Bickle A Chadwick H E Huppert M Hallworth and S Lyle Earth PlanetSci Lett 255 164 (2007)
39
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
questions about the environmental benefits of this processconcern the fate of the carbon dioxide over the sim104 yrperiod required for storage [2] Modelling the flow andretention of the carbon dioxide is beset by uncertaintiesThese relate to the physical structure of the reservoirs suchas permeability and stratal geometries [3] problemsinherent in modelling multi-phase flow [4] the behaviourof sealing strata in contact with CO2-rich fluids [5] thepossibility of reactions between CO2 and minerals in thereservoir [6] and the rate of progressive dissolution ofCO2 in the saline fluid filling the reservoirs [7] Theinformation necessary for robust a priori predictions ofthese phenomena is likely to be inadequate and much ofour understanding of the behaviour of CO2 storagereservoirs will necessarily be based on observations on
sites at which CO2 is currently being injected Mostpublished models of the movement of CO2 in geologicalreservoirs are based on numerical solutions [89] in whichparameters are adjusted to match the known history of thereservoir before the model is used to predict futurebehaviour (history matching [3]) The number of poorlyconstrained parameters limited data and the problems ofnumerical dispersion make it difficult to test theapplicability of assumptions inherent in the modelling
In this paper we model flow of CO2 at the Sleipnerstorage site in the North Sea by using modifications ofwell known solutions for gravity flows within a perme-able medium [eg 10] for an axisymmetric geometry[11] This straightforward but powerful analytical ap-proach best illustrates the controlling physics revealspredictive aspects of the behaviour of the CO2 andallows estimation of key reservoir parameters on thescale of the CO2 plume
In the Sleipner field about 8 million tons of CO2
have been injected since 1996 into the Utsira Sand asim200 m thick saline aquifer At the reservoir conditionsthe CO2 is less dense than the saline fluid filling thereservoir and rises buoyantly being partially trappedand ponded beneath a number of thin relatively im-permeable mudstone layers before reaching the muchthicker caprock overlying the Utsira Sand [12ndash14](Fig 1) Development of a prominent plume compris-ing distinct layers of CO2-saturated rock has beentracked by time-lapse seismic surveys in 1999 20012002 [1213] We use the seismic images to map theincrease in radius and thickness variations of theindividual CO2 layers with time [cf 13] (Fig 2)
Fig 1 Schematic illustration of CO2 injection at Sleipner and risingCO2 plumes being partially trapped under thin mudstones beforereaching Nordland Shale cap rock Note the vertical exaggeration
Fig 2 Seismic reflection profiles in 1994 1999 2001 and 2002 The 1994 pre-injection profile shows the base and top of the Utsira Sand but littledetail within the reservoir The subsequent post-injection profiles show bright reflections where CO2 is ponding under thin mudstones Note thepushdown of the basal Utsira Sand reflection resulting from low velocity of CO2 in the reservoir and development of a low amplitude verticallsquochimneyrsquo just to left (south) of the injection point (IP) presumed to be the main vertical conduit of CO2 in the plume [15] Layers are numbered in2002 profile Vertical scale is in two-way travel time
165M Bickle et al Earth and Planetary Science Letters 255 (2007) 164ndash176
Figure 12 Schematic cross-section through the Utsira sand showing the rising of carbondioxide plumes [1]
[2] R A Wooding J Fluid Mech 15 527 (1963)
410 Dislocation Structures in Microcantilevers
Supervisors Dr Cameron Hall Dr Ed Tarleton (Materials)Contact hallmathsoxacuk
When subjected to suitably large stresses metals stop behaving like perfectly elastic ma-terials and instead undergo ldquoplastic deformationrdquo leading to permanent shape changesAt a fundamental level plastic deformation of metals happens because of the productionand motion of crystal defects called dislocations While dislocations are hard to observeand model in large samples researchers in the Oxford Department of Materials have beenusing experiments on microcantilevers (ldquodiving boardsrdquo of metal as small as 6 micronslong) to explore dislocation motion and metal plasticity in a controlled environment
The experimental work on microcantilevers has happened in conjunction with MATLAB-based simulations using finite-element modelling and dislocation dynamics simulationsto describe the motion of dislocations These numerical simulations have revealed someinteresting properties of microcantilevers that we would like to explore further usingmathematical techniques As the displacement applied to the microcantilever increasesdislocation sources are activated and the dislocations form soft pile-ups (like traffic jamsof dislocations) in the interior of the microcantilever If the initial dislocation sourcedensity is sufficiently high the stress from these ldquotraffic jamsrdquo prevent some dislocationsources from being activated leading to a banded dislocation structure in the cantileverIn contrast if the initial dislocation source density is low (which often happens if the mi-crocantilever is very small) few dislocations can be produced and the plastic deformationis much smaller
The aim of this project is to use mathematical analysis to improve our understanding of
40
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
the critical phenomena that control plasticity scaling laws in microcantilevers This willallow new insight into the ldquosmaller is strongerrdquo size effect observed in mechanical testswhere the critical stress for plasticity increases as the characteristic sample dimensionis reduced To achieve this we will use numerical experiments based on the existingsimulation code and asymptotic analysis of soft pile-ups and associated dislocationstructures Suitable students should have an interest in asymptotic analysis and somefamiliarity with MATLAB Some knowledge of solid mechanics is also desirable but notnecessary
Through the course of the project the student would use numerical experiments toexplore the properties of the cantilever experiment formulate hypotheses for the fun-damental mechanisms that underlie the observed slip spacings and force-displacementscaling laws and explore these hypotheses using asymptotic analysis of dislocation struc-tures At the end of this project we hope to have developed mathematical models thatare simpler than the full simulations but which help explain the ldquosmaller is strongerrdquoeffect observed in microcantilevers and the scaling laws associated with it
411 Pattern Formation in Axisymmetric Viscous Gravity CurrentsFlowing over a Porous Medium
Supervisor Prof John WettlauferContact wettlaufermathsoxacuk
Gravity currents are primarily horizontal uid ows driven by a density dierence betweenthe intruding and ambient uids These ows are common in natural systems and in-dustrial processes and describe for example the spread of cold air into a room thedispersal of pollutants from an industrial spill and the ow of snow and debris avalanches(Huppert 2006) Many studies have examined in detail the propagation of currentsalong impermeable boundaries The topic here is to consider ow over porous substratesthrough which these currents can also drain It has been found experimentally thatwhen an axisymmetric viscous current flows of a porous medium of glass beads there area variety of unexpected behaviours including (1) overshoot of the steady-state radiusand subsequent retreat and (2) pattern formation in which axisymmetry appears to beviolated
The project will examine the sensitivity of patterns in experimental observations to fluid
41
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
properties and porous media geometry Under what conditions will axisymmetry be vio-lated When to scalloped patterns form and what controls their geometry The analysisof the existing theoretical model will focus on contact line pinning in the porous mediausing fluid mechanical methods such as asymptotic matching as well as experimentalmeasurements in the Mathematical Observatory
Expected outcomes involve new theoretical insights into this class of flows through anexplanation of the previous experimental results as well as those found during the projectGlycerin Currents
Glycerin input at corner of clear box filled with 3 mm diameter spheres
Non-axisymmetry
5 cm
Large scallop
Small scallops
5 cm
References
[1] H E Huppert Gravity currents a personal perspective J Fluid Mech 554 299ndash322 2006
[2] M J Spannuth J A Neufeld J S Wettlaufer and M G Worster Axisymmetricviscous gravity currents flowing over a porous medium J Fluid Mech 622 135ndash1442009
412 Finger Rafting The role of Spatial Inhomogeneity in PatternFormation in Elastic Instabilities
Supervisor Prof John Wettlaufer and Dr Dominic VellaContact wettlaufermathsoxacuk and vellamathsoxacuk
When two elastic sheets oating on a liquid collide intuition leads us to expect one of tworesults one sheet might be ldquosubductedrdquo under the other (as is observed in the Earthscrust) or the two might crush each other forming either a ridge line or a eld of rubble(as observed in thick ice oes) Observations of sea and lake ice reveal that there is infact a third possibility the formation of a series of interlocking ngers that alternatelyride over and under one another The ngers that are formed in this way are strikingprimarily because of their rectilinear features and the regular nger spacing
42
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
Under the assumption of spatially uniform physical properties this regime of behav-ior has been analyzed theoretically When this assumption is no longer valid we askhow does the regime diagram change The approach proceeds by rederiving the thinplate equations of a floating elastic sheet to embody a vertical profile in the mechanicalproperties and then solving the resulting problem of how the deformation responds toa point force The predictions will be tested with experimental measurements in theMathematical Observatory
Expected outcomes will be a more realistic theory of how plates on Earth can formpatterns during subduction and applications toward micromachining
Figure 13 Finger rafting of thin sea ice in the Amundsen Sea Photograph courtesy ofWilford F Weeks
References
[1] D Vella and J S Wettlaufer Finger Rafting A generic instability in floating elasticsheets Phys Rev Lett 98 088303 2007
[2] D Vella and J S Wettlaufer Explaining the patterns formed by ice floe interactionsJ Geophys Res 113 C11011 2008
43
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
5 Networks
51 Multilayer Networks
Supervisors Dr Mason Porter and Dr Mikko KivelaContact portermmathsoxacuk
Background and Problem Statement Most real and engineered systems includemultiple subsystems and layers of connectivity (see Figure 14 for an example) and itis important to take such features into account to try to improve our understandingof these systems It is thus necessary to generalize ldquotraditionalrdquo network theory mdash inwhich nodes are connected to each other via a single type of static edge mdash by developing(and validating) a framework and associated tools to study multilayer systems in acomprehensive fashion The origins of such efforts date back several decades and arosein multiple disciplines and now the study of multilayer networks has become one of themost important directions in network science This project will entail studying a problemto be determined based on student interest using the idea of multilayer networks
Figure 14 Multiple subsystems and layers of connectivity
Approaches and Techniques The particular approach and technique depends some-what on the particular problem but application-driven method-driven (and theory-driven) or data-driven projects are also possible Either one will include scientific com-putation and methods from network science will also of course be important in eithercase A tensorial formalism will be useful in many cases See Ref [1] below for somespecific examples of potentially relevant techniques which include multilinear algebrabranching processes dynamical systems perturbation theory mean-field analysis and
44
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
spectral theory Again the specific techniques will depend on the specific problem to bepursued
Hoped achievements The development of techniques to study multilayer networksis in its infancy and we hope to contribute to this exciting endeavor through the inves-tigation of a nice example (whether theory-driven and method-driven data-driven orapplication-driven)
Reference
[1] Kivela Mikko Arenas Alexandre Barthelemy Marc Gleeson James P MorenoYamir and Porter Mason A [2013] Multilayer Networks submitted to Journal ofComplex Networks httparxivorgabs13097233
(Note [1] is currently the only review article on the topic The references therein alsocontain a lot of additional information and examples of specific problems that have beenstudied)
52 Computational Topology for Neuroscience
Supervisors Dr Heather Harrington and Dr Mason PorterContact harringtonmathsoxacuk and portermmathsoxacuk
Background and Problem Statement The nervous system is responsible for sensoryand motor skills via neurons glia and synapses In the brain neurons process andtransmit information of cellular signals The set of neurons and the connections betweenthem mdash called the connectome mdash can be described as a network and it has thus beenstudied using ideas from subjects like network science For example the nodes of astructural neuronal network are the neurons which are connected by edges that representsynapses By contrast in functional neuronal networks cortical areas yield the networknodes which are connected to each other via behavioural similarity (eg coherence intheir firing patterns) One obtains data to construct such networks via neuroimagingtechniques and network analysis of brain networks has become increasingly popularHowever many challenges remain in the quest to understand the global behaviour offunctional neuronal networks
Description of the approach planned and the teachniques needed Neuronalnetworks have also been studied using computational topology whereby one applies al-gorithmic methods for understanding shapes and complexes in data structures such aspoint clouds or networks A method known as persistent homology enables one to inves-tigate the properties of a space at multiple spatial resolutions This makes it possible toobtain information about structures in data that persist in some sense See for examplethe holes of different dimensions that persist in the network in the figure below Suchapproaches are powerful for the investigation of global low-dimensional structure withinnetworks and other high-dimensional data They have only been applied to functionalbrain networks in the last few years and researchers have only scratched the surface oftheir potential insights and utility in this area
45
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-
Plan to achieve The aim of this project is to successfully apply tools from com-putational topology to data from neuroscience experiments and the output of models(eg nonlinear oscillator models such as the Kuramoto model) The student will learnabout neuroscience networks and persistent homology The computational methodswill require learning about topological notions such as Betti numbers filtrations sim-plicial complexes and software for constructing persistence diagrams in networks andother data This project can be either data-driven or theory-driven and we expect toobtain interesting insights in both neuroscience and mathematical (and computational)methodology
References
[1] Bassett DS et al (2013) Chaos 23 013142
[2] Bullmore ET amp Sporns O (2009) Nat Rev Neurosci 10186ndash198
[3] Bullmore ET amp Bassett DS (2011) Annu Rev Clin Psychol 7(1)113ndash140
[4] Dabaghian Y et al (2012) PLoS Comput Biol 8(8)e1002581
[5] Petri G et al (2013) PLoS One 8(6)e66505
46
- Projects with the Industrial Sponsors of the MSc
-
- Sharp mdash Flow and Solidification in Confined Geometries with Industrial Applications
-
- Numerical Analysis Projects
-
- Chebfun Dissertation Topics
- Multi-Structures and Computing in Mixed Dimensions
- Parallel Computing for ODEsPDEs with Constraints
- Segmentation and Registration of Lung Images
- Repairing Damaged Volumetric Data using Fast 3D Inpainting
- Constraints and Variational Problems in the Closest Point Method
- Topics in Matrix Completion and Dimensionality Reduction for Low Rank Approximation
- Optimisation of Tidal Turbines for Renewable Energy
- Uncertainty Quantification in Glaciological Inverse Problems
- Edge Source Modelling for Diffraction by Impedance Wedges
- What to do with DLA
- Random Plane Wave and Percolation
- Numerical Solution of Equations in Biochemistry
- Numerical Solution of the Rotating Disc Electrode Problem
-
- Biological and Medical Application Projects
-
- Circadian Rhythms and their Robustness to Noise
- The Analysis of Low Dimensional Plankton Models
- Individual and Population-Level Models for Cell Biology Processes
- A New Model for the Establishment of Morphogen Gradients
- Modelling the Regrowth and Homoeostasis of Skin
- Modelling the Growth of Tumour Spheroids
- DiscreteHybrid Modelling of Lymphangiogenesis
- Mathematical Modelling of the Negative Selection of T Cells in the Thymus
- The Dynamics and Mechanics of The Eukaryotic Axoneme
-
- Physical Application Projects
-
- Swarm Robotics From Experiments to Mathematical Models
- A Simple Model for Dansgaard-Oeschger Events
- Modelling Snow and Ice Melt
- A Network-Based Computational Approach to Erosion Modelling
- Retracting Rims
- Modelling Spray Deposition for Applications in Manufacturing Supercapacitors
- Mathematical Modelling of Membrane Fouling for Water Filtration
- Flow-Induced ``Snap-Through
- Plumes with Buoyancy Reversal
- Dislocation Structures in Microcantilevers
- Pattern Formation in Axisymmetric Viscous Gravity Currents Flowing over a Porous Medium
- Finger Rafting The role of Spatial Inhomogeneity in Pattern Formation in Elastic Instabilities
-
- Networks
-
- Multilayer Networks
- Computational Topology for Neuroscience
-