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Study of Immiscible Liquid-Liquid Microfluidic Flow using
SPH-based Explicit Numerical Simulation
Hamideh Elekaei Behjati
A thesis submitted for the degree of Doctor of Philosophy
School of Chemical Engineering
The University of Adelaide
Australia
November 2015
i
Declaration
I certify that this work contains no material which has been accepted for the award of any
other degree or diploma in my name, in any university or other tertiary institution and, to
the best of my knowledge and belief, contains no material previously published or written
by another person, except where due reference has been made in the text. In addition, I
certify that no part of this work will, in the future, be used in a submission in my name, for
any other degree or diploma in any university or other tertiary institution without the prior
approval of the University of Adelaide and where applicable, any partner institution
responsible for the joint-award of this degree.
I give consent to this copy of my thesis when deposited in the University Library, being
made available for loan and photocopying, subject to the provisions of the Copyright Act
1968. The author acknowledges that copyright of published works contained within this
thesis resides with the copyright holder(s) of those works.
I also give permission for the digital version of my thesis to be made available on the web,
via the University’s digital research repository, the Library Search and also through web
search engines, unless permission has been granted by the University to restrict access for
a period of time.
Hamideh Elekaei Behjati
22 November 2015
ii
Abstract
Microfluidic devices are utilized in a wide range of applications, including micro-
electromechanical devices, drug delivery, biological diagnostics and micro-fuel cell
systems. Of particular interest here are liquid-liquid microfluidic systems; which are used
in drug discovery, food and oil industry amongst others. Increased understanding of the
fundamentals of flows in such devices and an improved capacity to design them can come
from modelling. In the case of liquid-liquid flows in microfluidic systems, it is necessary
to explicitly model the behaviour of the individual liquid phases. Such explicit numerical
simulation (ENS) as it is termed requires advanced numerical methods that are able to
evaluate flow involving multiple deforming fluid domains within often complex
boundaries. Smoothed Particle Hydrodynamics (SPH), a Lagrangian meshless method, is
particularly suitable for such problems. This use of a CFD allows determination of
parameters that are difficult to determine experimentally because of the challenges faced in
microfabrication. The study reported in this thesis addresses these concerns through
development of a new SPH-based model to correctly capture the immiscible liquid-liquid
interfaces in general and for a microfluidic hydrodynamic focusing system in particular.
The model includes surface tension to enforce immiscibility between different liquids
based on a new immiscibility model, enforces strict incompressibility, and allows for
arbitrary fluid constitutive models. This work presents a detailed study on the effects of
various flow parameters including flowrate ratio, viscosity ratio and capillary number of
each liquid phase, and geometry characteristics such as channel size, width ratio, and the
angle between the inlet main and side channels on the flow dynamics and topological
changes of the multiphase microfluidic system. According to our findings, both flowrate
quantity and flowrate ratio affect the droplet length in the dripping regime and a large
viscosity ratio imposes an increase in the flowrate of the continuous phase with the same
capillary number of the dispersed phase to attain dripping regime in the outlet channel.
Also, increasing the side channel width causes longer droplets, and the right-angled design
makes the most efficient focusing behaviour. This study will provide great insights in
designing microfluidic devices involving immiscible liquid-liquid flows.
iii
Achievements
Three following papers were achieved from this work:
1) Elekaei Behjati, H., Navvab Kashani, M., Biggs, M. J., Modelling of Immiscible
Liquid-Liquid Systems by Smoothed Particle Hydrodynamics, to be submitted to
Computational Mechanics, 2015.
2) Elekaei Behjati, H., Navvab Kashani, M., Zhnag, H., Biggs, M. J., Smoothed
Particle Hydrodynamics-based Simulations of Immiscible Liquid-Liquid
Microfluidics in a Hydrodynamic Focusing Configuration, to be submitted to
Physics of Fluids, 2015.
3) Elekaei Behjati, H., Navvab Kashani, M., Zhnag, H., Biggs, M. J., Effects of Flow
Parameters and Geometrical Characteristics on the Dynamics of Two Immiscible
Liquids in a Hydrodynamic Focusing Microgeometry, to be submitted to Physics
of Fluids, 2015.
iv
Acknowledgements
Over the past four years, I was inspired and supported by many people without whom I
would never have been able to finalize this dissertation. This section is therefore dedicated
to all those people who have shared their time, enthusiasm and expertise with me,
admitting that I will never be able to thank enough those whom I am indebted the most. I
would like to express my deepest gratitude and appreciation to all of them.
I would like to start by thanking my principal supervisor, Prof Mark J. Biggs for his
ongoing advice and support during the entire course of my PhD study. Mark, your
expertise and detailed critical comments on every part of my research, pushed me to think
deeper about my research and shape my ideas. I am also very grateful to the other members
of the supervision team, A/Prof Zeyad Alwahabi and Dr Hu Zhang for their valuable
feedback and supports.
During the four years of my PhD, I have been lucky to discuss my research and collaborate
with many bright people. In particular, I wish to extend my appreciation to Dr Milan
Mijajlovic. Milan, thanks for your occasional support in computer coding and
programming.
I am more thankful than I can say to Mrs Jacqueline Cookes, Mrs Marilyn Seidel and Mr
Ron Seidel. Mrs Jacqui, you have been always nice and kind to me, that made me feel at
home. You have been a cheerful caring friend for me. Your honesty, experience and
kindness make you a person with whom talking is nothing else but a pleasure. I have learnt
a lot from you that will never forget. Marilyn and Ron, I am not able to describe your big
hearts in a few words, I can just wholeheartedly thank you both for all the endless support I
have received from you. You all made my PhD journey much more enjoyable. Thanks for
everything.
The Government of Australia and the University of Adelaide are gratefully acknowledged
for providing me Adelaide Scholarships International (ASI) for my study in Australia. The
cluster super-computational facilities and high-performance computing (HPC) system for
this work were provided by eResearch SA organization, which is warmly appreciated.
v
I am deeply indebted to my loving parents, who have always been encouraging me to
continue my education and broaden my mind. Dad, thank you for your infinite support.
Mum, you created a home environment that encouraged me to study and learn. Words
cannot describe how grateful I am for your never ending love and care and all you have
done for me. Without your support, I would never achieve what I have today. I am also
thankful to my caring brothers, Majid, Habib and Mehdi, and my lovely sisters, Zahra and
Shabnam, for all their support, accompany, and encouragement during my PhD study
overseas. Likewise, I would like to thank my parents-in-law for their support, inspiration
and influence.
Finally, and above all, I cannot begin to express my unfailing gratitude and love to my best
friend, my dear husband, Dr Moein N. Kashani. Moein, you have always supported me
through every step of this long journey, and have constantly encouraged me when the tasks
seemed arduous and insurmountable. Without you, I would never have had the inspiration
and motivation to complete this dissertation. Thank you for your warm presence in my life.
Hamideh Elekaei Behjati
November 2015
vi
List of Figures
Figure 2-1 Examples of liquid-liquid microfluidics; (a) integrated microfluidic liquid-
liquid extraction [92] and (b) water-in-oil-in-water (W/O/W) emulsion in microchannels
[93] .........................................................................................................................................9
Figure 2-2 Possible flow patterns in immiscible-liquid microfluidics at (a) cross-junction
[14], the flow regimes from top to bottom show threading, jetting, dripping, tubing, and
viscos displacement regimes, (b) T-junction [94], the flow patterns from top to bottom
show slugs, monodispersed droplet, droplet populations, parallel flows, parallel flows with
wavy interface, chaotic thin striations flow .........................................................................10
Figure 2-3 Typical Ca-based flow map for immiscible-liquid two-phase microfluidics [14]
..............................................................................................................................................11
Figure 2-4 Computational mesh close to the interface for a rising droplet [159] and (b)
adaptive mesh refinement for a droplet impact on a Liquid-Liquid interface [160] ............18
Figure 3-1 An illustration of an SPH weighting function with compact support sited on
particle-i (shown in red) that leads to the particle interacting with all other particles-j
within the support of size O(h). ...........................................................................................35
Figure 3-2 Typical variation through time, 𝒕 ∗= 𝒕𝜸𝝆𝒉𝟑, of the circumference of an
initially square neutrally buoyant droplet suspended in a second continuous phase and the
associated change in pressure difference, ∆𝑷 ∗= ∆𝑷𝒉𝜸, across the interface between the
two liquids. Snapshots of the droplet along the transformation pathway are shown at
various points; the continuous phase is not shown for simplicity. .......................................41
Figure 3-3 Variation of dimensionless pressure drop across the droplet interface with the
inverse of its dimensionless radius, 𝑹 ∗= 𝑹𝒉, for 𝜸 = 𝟎. 𝟎𝟏𝟒𝟐 N/m (obtained using
𝜺 = 𝟏𝟔 N/m2). ......................................................................................................................42
Figure 3-4 Variation of the interfacial tension, 𝜸, with the interaction model parameter that
induces immiscibility here, 𝜺. ..............................................................................................42
Figure 3-5 Droplet shape and velocity field obtained from a typical SPH simulation with
the key characteristics shown. ..............................................................................................43
Figure 3-6 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟓,
𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫𝑯 = 𝟎. 𝟓. .......................................................................................44
vii
Figure 3-7 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟐,
𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫𝑯 = 𝟎. 𝟓. .......................................................................................45
Figure 3-8 Variation of the deformation parameter as defined in Equation (24) with
Capillary Number as predicted by SPH with Np=7701 (solid circles with linear fit shown
as a line; R^2=0.99), SPH with Np=4961 (open circles) Taylor theory as defined by
Equation (25) (solid line) [244], and experimental data (open squares) [245]. ...................46
Figure 3-9 Variation of the droplet tilt angle with Capillary Number as predicted by SPH
with Np=4961(open circles), SPH with Np=7701 (solid circles), Taylor theory as defined
by Equation (25) (solid line) [244], and the experimental values of Sibillo [246] (open
squares). ...............................................................................................................................47
Figure 3-10 Variation of the droplet descent speed with time as predicted by SPH (dash
line) and Han and Tryggvason [247] (solid line) for 𝛈 = 𝟏. 𝟏𝟓, 𝛌 = 𝟏, 𝐄𝐨 = 𝟏𝟎, 𝐎𝐡𝒅 =
𝟎. 𝟐𝟒 and 𝐌𝐨 = 𝟎. 𝟎𝟒. Snapshots of the deforming droplet are shown at various times....48
Figure 3-11 (a) variation of droplet descent speed with time as predicted by SPH for
𝜼 = 𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒 (solid line), and 𝑬𝒐 = 𝟏𝟒𝟒 (dash line).
Droplet deformation for (a) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒, and (c) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑,
𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟏𝟒𝟒 . ...................................................................................................49
Figure 3-12 (a) variation of droplet descent speed with time as predicted by SPH for
𝜼 = 𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟏. 𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 (solid line), and 𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 (dash
line). Droplet deformation for 𝑶𝒉𝒅 = 𝟏. 𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, (b) 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 and 𝑴𝒐 =
𝟎. 𝟎𝟎𝟎𝟏𝟓, and (c) 𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 and 𝑴𝒐 = 𝟓𝟖 ...................................................................49
Figure 4-1 (a) schematic of cross-junction flow focusing geometry considered in detail in
[14] and here, where 𝑸𝒊, 𝑪𝒂𝒊, 𝝁𝒊 and 𝑼𝒊 = 𝑸𝒊𝑯𝟐 are the flow rate, Capillary Number,
viscosity and superficial velocity of liquid Li, respectively, and 𝜸𝟏𝟐 is the interfacial
tension for the liquid pair; (b) phase morphology map based on the Capillary Numbers of
the liquid L1 (𝑪𝒂𝟏) and the focusing liquid L2 (𝑪𝒂𝟐): (a) threading; (b) jetting; (c)
dripping; (d) tubing; and (e) viscous displacement. The numbers on the map refer to the
conditions that were considered in the study reported on here. ...........................................56
Figure 4-2 Representative outcome for threading regime at 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐,
which corresponds to point 1 on Figure 4-1(b). ...................................................................60
viii
Figure 4-3 Dimensionless thickness of threads as a function of flowrate ratio as evaluated
using the SPH model (open circles) and equation (1) (solid line). ......................................61
Figure 4-4 Representative outcome of the SPH model for dynamics of the droplet
formation in dripping regime at 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑, which corresponds to point
2 on Figure 4-1(b). The snapshots belong to (a) 𝐭 ∗= 𝟎. 𝟎𝟕 , (b) 𝐭 ∗= 𝟎. 𝟐 , (c) 𝐭 ∗= 𝟎. 𝟑𝟖 ,
and (d) 𝐭 ∗= 𝟎. 𝟒 , where 𝐭 ∗= 𝐔𝟏𝐭𝐇 ...................................................................................63
Figure 4-5 Dimensionless length of droplet as estimated by the SPH model (open circles)
and equation (2) (solid line) .................................................................................................63
Figure 4-6 Representative outcome of the SPH model for dynamics of the jetting regime at
𝑪𝒂𝟏 = 𝟎. 𝟑 and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟓, which corresponds to point 3 on Figure 4-1(b)). The
snapshots belong to (a) 𝒕 ∗= 𝟎. 𝟒𝟐 , and (b) 𝒕 ∗= 𝟎. 𝟒𝟒 , where 𝒕 ∗= 𝑼𝟏𝒕𝑯 .....................64
Figure 4-7 Dimensionless (a) droplet diameter (equation (3)), and (b) thread length
(equation (4)) in dripping regime as evaluated by the SPH model (open circles) and the
related correlations (solid line).............................................................................................65
Figure 4-8 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 = 𝟏. 𝟎
and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟔, which corresponds to point 4 on Figure 4-1(b). ...................................65
Figure 4-9 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 =
𝟎. 𝟏𝟗 and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟎𝟐, which corresponds to point 16 on Figure 4-1(b). ....................66
Figure 4-10 Representative outcome for threading regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 =
𝟎. 𝟎𝟎𝟖, which corresponds to point 12 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟒 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟏,
which corresponds to point 11 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟑 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐𝟓,
which corresponds to point 10 on Figure 4-1(b). .................................................................67
Figure 4-11 Representative outcome for dripping regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟖𝟓 and 𝐂𝐚𝟐 =
𝟎. 𝟎𝟎𝟐 which corresponds to point 15 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟏,
which corresponds to point 8 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟔
which corresponds to point 7 on Figure 4-1(b). ...................................................................68
Figure 4-12 Representative outcome for jetting regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟏𝟗 and 𝐂𝐚𝟐 =
𝟎. 𝟎𝟎𝟓 which corresponds to point 15 on Figure 1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟏𝟕 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖,
which corresponds to point 14 on Figure 1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟐 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖
which corresponds to point 13 on Figure 1(b). ....................................................................69
Figure 5-1 Schematic of cross-junction flow focusing geometry ........................................76
ix
Figure 5-2 Effect of flowrate ratio on the flow pattern at (a) q=0.1, (b) q=0.5, (c) q=1, (d)
q=2, (e) q=3, (f) q=5, (g) q=10, and (h) q=20, with a fixed viscosity ratio of 𝛍𝟏𝛍𝟐 = 𝟗𝟒78
Figure 5-3 Flow regime map in terms of 𝐂𝐚𝟏, 𝐂𝐚𝟐 and flowrate ratio. Dripping (solid
square), displacement (solid circles), tubing (solid triangle), threading (solid diamond), and
jetting flow regime (solid pentagon). ...................................................................................79
Figure 5-4 Effect of the flowrate quantities on the droplet size for the constant flowrate
ratio of q=2 and at (a) 𝐐𝟏 = 𝟏𝟎 µ𝐥𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟖𝟔 , (b) 𝐐𝟏 =
𝟐𝟓 µ𝐥𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟏, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟐𝟐 and (c) 𝐐𝟏 = 𝟒𝟎 µ𝐥𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟏𝟔𝟖,
𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑𝟔 , (d) dimensionless droplet diameter vs. flowrate of 𝐐𝟏 .........................80
Figure 5-5 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0036, (b) λ=264 and
Ca2=0.02, and (c) λ=24 and Ca2=0.2, on flow pattern at q=10 and Ca1=0.5 .......................81
Figure 5-6 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.00054, (b) λ=264 and
Ca2=0.003, and (c) λ=24 and Ca2=0.033 on flow pattern at q=8 and Ca1=0.1 ....................82
Figure 5-7 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0009, (b) λ=264 and
Ca2=0.005 and (c) λ=24 and Ca2=0.054, on flow pattern at q=4.4 and Ca1=0.3 .................83
Figure 5-8 Effect of width ratio of (a) 𝛇 = 𝟎. 𝟓 and Ca2=0.0036, (b) 𝛇 =1 and Ca2=0.0009,
and (c) 𝛇 =2 and Ca2=0.00026 on droplet length at 𝐪 = 𝟐 and 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒 ...................84
Figure 5-9 Normalized droplet length in terms of the width ratio 𝛇 and the flowrate ratio q,
SPH simulation (solid diamond) and Equation (1) (open diamond) for𝛇 = 𝟎. 𝟓, SPH
simulation (solid circle) and Equation (1) (open circle) for 𝛇 = 𝟏, and SPH simulation
(solid square) and Equation (1) (open square) for 𝛇 = 𝟐 .....................................................85
Figure 5-10 Effect of an inlet angle of (a) θ=45⁰, (b) θ=90⁰ and (c) θ=135⁰ on the droplet
length at q=2, Ca1=0.1 and Ca2=0.002 .................................................................................86
x
List of Tables
Table 4-1: Properties of liquid pair considered ....................................................................59
xi
Abbreviations
Ca Capillary Number
CAD Computer Aided Design
CFD Computational Fluid Dynamics
DNS Direct Numerical Simulation
DPD Dissipative Particle Dynamics
ENS Explicit Numerical Simulation
Eo Eotvos Number
FDM Finite Difference Method
FEM Finite Element Method
FVM Finite Volume Method
HPC High-Performance Computing
LBM Lattice Boltzmann Method
LGA Lattice Gas Automata
LOC Lab-On-a-Chip
LoG Laplacian of Gaussian
LSM Level Set Method
MD Molecular Dynamics
Mo Morton Number
NNPS Nearest Neighbouring Particle Searching
Oh Ohnesorge Number
PPE Pressure Poisson Equation
Re Reynolds Number
SD Standard deviation
SPH Smoothed Particle Hydrodynamics
VOF Volume of Fluid
µTAS Micro Total Analysis System
3D Three-dimensional
2D Two-dimensional
xii
Nomenclatures
a [m] Length of the major axis of the ellipsoid
A [m2] Area
b [m] Length of the minor axis of the ellipsoid
c [ - ] Colour function
CD [ - ] Drag coefficient
g [m/s2] Gravitational acceleration
h [m] Cut off distance
I [ - ] Unit tensor
L0 [m] Initial distance between particles
m [kg] Mass of particle
Np [ - ] Number of SPH particles
P [Pa] Pressure
∆𝑃 [Pa] Pressure drop
r [m] Position
R [m] Radius of droplet
𝑟𝑖𝑗 [m] Distance between particles i and 𝑗
t [s] Time
t [ - ] Time step
∆𝑡 [s] Time step size
u [m/s] Volume averaged fluid velocity
U [m/s] Superficial velocity
𝑈𝑚𝑎𝑥 [m/s] Maximum characteristic velocity
V [m3] Particle volume
V [m/s] Velocity vector
𝑊 [m-3
] Smoothing kernel
x [m] Distance vector
[ - ] Interfacial tension
[ - ] Dirac delta function
[ - ] Thread thickness
𝜀 [ - ] Immiscibility model parameter
[ - ] Density ratio
λ [ - ] Viscosity ratio
µ [Pa.s] Dynamic viscosity
𝜌 [kg/m3] Density
σ N/m2 Stress
τ N/m2 Shear stress
𝜉 [ - ] Channel width ratio
Gradient operator
xiii
Subscript
c Value for continuous phase
i Value for particle of interest
j Value for neighbouring particles
j Value for jetting
d Value for droplet
p Value for particle
w Related to wall
α -coordinate direction
β β-coordinate direction
* Intermediate state
Superscript
α Number of dimensions
β Number of dimensions
* Dimensionless value
xiv
Table of contents
Declaration…. ....................................................................................................................... i
Abstract……. ....................................................................................................................... ii
Achievements ...................................................................................................................... iii
Acknowledgements ............................................................................................................. iv
List of Figures ..................................................................................................................... vi
List of Tables ........................................................................................................................x
Abbreviations ..................................................................................................................... xi
Nomenclatures ................................................................................................................... xii
Table of contents .............................................................................................................. xiv
Introduction ....................................................................................................1 Chapter 1:
Literature Review...........................................................................................6 Chapter 2:
2.1 Microfluidics ................................................................................................................6
A brief history of microfluidics .............................................................................6 2.1.1
Advanced techniques to make microfluidics .........................................................7 2.1.2
Brief overview of current microfluidic flows ........................................................7 2.1.3
2.2 Multiphase microfluidics ..............................................................................................8
2.3 Immiscible Liquid-Liquid Microfluidics ......................................................................8
Experimental study of immiscible liquid-liquid microfluidic system .................12 2.3.1
2.4 Numerical study of immiscible liquid-liquid microfluidic system ............................15
2.5 Smoothed Particle Hydrodynamics (SPH) Method ....................................................21
General SPH Formulation ....................................................................................23 2.5.1
Neighbour searching algorithm ...........................................................................25 2.5.2
SPH Algorithm ....................................................................................................26 2.5.3
2.6 Conclusion ..................................................................................................................26
Modelling of Immiscible Liquid-Liquid Systems by Smoothed Particle Chapter 3:
Hydrodynamics ..................................................................................................................28
3.1 Introduction ................................................................................................................31
3.2 The Method ................................................................................................................34
Governing Equations ...........................................................................................34 3.2.1
xv
SPH Discretization ...............................................................................................35 3.2.2
Immiscibility Model ............................................................................................37 3.2.3
Solution Algorithm ..............................................................................................38 3.2.4
3.3 Results and Discussion ...............................................................................................40
Young-Laplace and Parameterization of Immiscibility Model ...........................40 3.3.1
Deformation of a Confined Droplet in a Linear Shear Field ...............................42 3.3.2
Deformation of Free-Falling Droplet ...................................................................47 3.3.3
3.4 Conclusions ................................................................................................................50
Smoothed Particle Hydrodynamics-based Simulations of Immiscible Chapter 4:
Liquid-Liquid Microfluidics in a Hydrodynamic Focusing Configuration ..................51
4.1 Introduction ................................................................................................................54
4.2 Model and Study Details ............................................................................................58
Model ...................................................................................................................58 4.2.1
Study Details ........................................................................................................59 4.2.2
4.3 Results and Discussion ...............................................................................................59
4.4 Conclusion ..................................................................................................................66
4.5 Supplementary Information ........................................................................................67
Effects of Flow Parameters and Geometrical Characteristics on the Chapter 5:
Dynamics of Two Immiscible Liquids in a Hydrodynamic Focusing Microgeometry 70
5.1 Introduction ................................................................................................................73
5.2 Model Details .............................................................................................................75
5.3 Results and Discussion ...............................................................................................76
Effects of flowrate and flowrate ratio ..................................................................76 5.3.1
The effect of the viscosity ratio ...........................................................................80 5.3.2
Effect of geometry ...............................................................................................83 5.3.3
5.3.3.1 Width ratio effect ..........................................................................................83
5.3.3.2 Inlet angle effect ............................................................................................85
5.4 Conclusion ..................................................................................................................86
Conclusion .....................................................................................................88 Chapter 6:
References… .......................................................................................................................90
1
Introduction Chapter 1:
Microfluidics, the study of fluid motion at a micro-scale, is currently a growing research
area. It has practical applications in the design of systems in which a small amount of fluid
in channels with dimensions in the order of tens to hundreds of micrometres is manipulated
and processed [1-3]. Increased understandings and developments in microfluidics will lead
to drastic progress in biotechnology, drug discovery, microelectronics, energy generation,
pharmaceutical, food and many other industries. For instance, microfluidics allows the
breakup of a droplet of blood into thousands of micro-droplets to perform biological tests
in parallel [3] , and this method is far better than those current ones, which require much
larger samples and have slower reaction times.
At micro-scale, a high surface area to volume ratio results in that the interfacial forces,
such as viscous force and surface tension, become dominant and intensely influences the
fluid flow behaviour [4]. The advancement in micro-engineering and the scale-dependence
behaviour of fluids allow us to be able to accurately control multiple fluid interfaces,
which have been utilised in different areas ranging from chemical processing and energy to
medical science and biology [4]. Practically, microfluidics offers a wealth of advantages
including accurate manufacturing of devices, and precise control of interfaces.
Microfluidic technology allows us to manipulate the deformable and moving interfaces in
multiphase systems to achieve the benefits which cannot be obtained from the macro-scale
flow devices.
Of particular interest are such multiphase microfluidic systems, such as micro-extraction
used in drug discovery [5, 6], and micro-emulsification in food, cosmetic, pharmaceutical
and oil industries [7-12]. The multiphase flow and moving/deformed interfaces
unfortunately are far less well developed compared to the single-phase counterpart.
Although a variety of experimental studies have been done so far on the droplet-based
microfluidics [13-21], they are still sporadic, because of only limited material used for
fabrication, a small range of flow parameters, and few configurations and geometries
tested during measurement. Importantly, these reports are more focused on the observed
phenomena, and the physics for such phenomena is not well clarified. Therefore, a full
2
understanding of the fundamental physics of multiphase liquid flow dynamics and the
effects of geometry in the microfluidics and process parameters on the moving interfaces
are essentially needed. Experimental studies for microchannel multiphase flows are quite
challenging and expensive, while analytical solutions at such small scales are limited to
very few simple cases. Consequently, modelling and numerical studies are often required
to address complex flows, which are poorly understood, and to facilitate developments in
microfluidics. To this end, this project aims to develop a numerical model for the
immiscible liquid-liquid phase in microfluidics, which can be used as foundation for
building a design capability for multichannel microfluidics devices.
Due to complex features of multiphase flow in microchannel devices including moving
interfaces, a large ratio of the surface to volume and micro-mechanics such as capillary
forces and potential intricate boundaries (e.g. fluid-fluid interface dynamics), numerical
simulation of multiphase flows in microfluidics is not straightforward. Computational
Fluid Dynamics (CFD) has been extensively used to simulate fluid flows in micro-fluidic
devices as indicated by literature [22, 23]. These have been, however, largely attributed to
a single phase systems. In the case of a multiphase system, the dearth of studies is due to
challenges in explicitly tracking the individual phases. It is not possible to simply adopt a
multi-fluid modelling approach for the immiscible-liquid multiphase microfluidic system,
which treats the immiscible liquids as a homogeneous mixture without any interfaces and
is typically applied for macro-scale counterparts [24].
Explicit numerical simulation (ENS) is a general concept where one attempts to model all
the significant phenomena explicitly rather than in a mean field or average way as is done
in many engineering models; in this sense, direct numerical simulation (DNS) of
turbulence is a class of ENS [25]. Explicit simulation of the multiphase system allows us to
track individual phases, deal with complex phase geometries and topologies, and consider
multiple time and/or length scales in one simulation [26]. In most of the cases in
microchannels, it is become important to explicitly model the behaviour of the individual
phases and the way they have interaction in multiphase flow. In addition to that, based on
the application of microfluidic devices, we are interested in the shape of dispersed phase
rather than the other things. It means that, in addition to the continuous fluid phase,
disperse phase must be explicitly modelled.
3
In the explicit simulation, modelling and tracking the interface with large deformation and
topological changes is the main challenge of traditional CFD methods which need high
resolution meshes, mesh refinement, proper mesh aspect ratios and intricate discretization
techniques. Among those challenging issues, mesh generation and regeneration is a
problematic task. It is a time-consuming and computationally-expensive process for
problems with a complex geometry. Constructing a regular mesh for an irregular or
intricate geometry may carry more computational load than solving the problem itself.
Equally challenging tasks are to determining accurate locations of the free surfaces,
deformable boundaries and moving interfaces within the frame of the fixed grid. This
mesh-based numerical simulation is an expensive operation, which also bears another
source for modelling errors and numerical instability [27-30]. Therefore, simulation of
multiphase microfluidic systems using mesh-based ENS methods is computationally
expensive and inefficient. Therefore, algorithms based on the Lagrangian meshless method
should be considered in these problems.
Meshless methods have been developed over the past few decades with more efficient
computational schemes designed for more complicated problems. Regarding dynamic
simulations of continuum materials experiencing large deformations, the mesh creation
processes in conventional mesh-based methods are involved by inevitability re-meshing
the domain with the intention to keep away from weaken accuracy of the results owing to
reduced aspect ratios of the mesh and termination of the simulation because of mesh
complication and inversion. In this context, meshless particle methods have no inherent
restrictions on the extent of deformation of the considered domain, since the connectivity
between particles is updated for each computation step. Therefore, particle method holds
advantages for large deformation simulations as mesh-based methods require constant re-
meshing to obtain well-behaved elements. Moreover, the method is capable to be
connected with a CAD database much simpler than mesh-based approaches due to the fact
that this method does not need to generate an element grid. Meshless particle methods also
produce more accurate results. Discretization techniques in meshless particle methods
allow prices depiction of geometric object in compared with traditional mesh-based
methods [27, 31, 32].
4
One preferable meshless particle method is Smoothed Particle Hydrodynamics (SPH) [33]
which is a fully Lagrangian meshless method. For computational fluid dynamics problems
governed by the Navier-Stokes equations, SPH is a powerful solution technique. In the
SPH method, a set of particles that have material properties and interact with each other
under the area delimited by a weight function or smoothing function, demonstrate the state
of a system [34]. With the aid of these discrete particles, the governing equations are
discretised and a diversity of particle-based formulations is used to calculate the
acceleration, velocity and local density of the fluid. Afterwards through an appropriate
equation of state, fluid pressure is calculated and hence the particle acceleration will be
achieved using the pressure gradient and density. Since the movement of the fluid is
illustrated by the movement of the particles, there is no extra procedure for interface
tracking in multiphase flows [35].
Compared to the long-established mesh-based numerical methods, SPH has some
particular benefits. Firstly, SPH is a meshless particle method with Lagrangian nature,
which can record temporal profile of the material particles and so flow of the system can
be accordingly obtained. Secondly, by positioning appropriate particles at particular
locations at the early stage in the process of the simulation, free surfaces, interfaces and
moving boundaries can be traced. As a consequence, interface tracking is explicit through
capturing the locations of the particles, much simpler than Eulerian methods. Therefore,
SPH is an ideal candidate for modelling free surface and interfacial flow problems.
Thirdly, SPH is also appropriate for situations where the object of interest is not a
continuum, such as micro-scale and nano-scale simulation in bio-engineering and nano-
engineering. In addition, SPH is reasonably less complicated in numerical implementation,
especially in comparison with mesh-based methods. Therefore, the SPH will be chosen to
apply to the immiscible liquid-liquid flow in microchannel system in this project.
There are only a limited number of numerical studies with regard to immiscible-liquid
multiphase microfluidics in literature [23, 35-37]. Among them, lattice Boltzmann method
(LBM) has been mostly employed for simulation of droplet-based microfluidic systems
[32, 38, 39], however, LBM-based simulation suffer from stability problems [40].
This project will be one of the first to apply SPH, a powerful meshless particle method, to
explicitly modelling immiscible liquid-liquid flows in microchannels. This novel
5
modelling will enable us to better understand the effect of flow parameters, microfluidic
geometry, and interfacial phenomena in such microfluidic devices. Moreover, this work
will provide great insights in designing microfluidic devices involving immiscible liquid-
liquid flows.
The aim of the work reported in this thesis was directed towards developing the application
of meshless method to the explicit numerical simulation of immiscible- liquid flows in
microfluidic systems so as to achieve an increased understanding of fluid dynamics and
phase morphologies within the phases in such systems. The study reported in this thesis
addresses these concerns through two related developments. ENS of immiscible liquid-
liquid flows based on incompressible SPH method with particular regard to correctly
capturing the liquid-liquid interface represents the first development. This goal was
achieved through modelling of the system, code development, and validation of the model
by applying it to a range of interfacial problems (Objective I). The second development of
the work has been divided into two parts: the first involved the application of the proposed
model for the study of hydrodynamic focusing of a liquid flowing in a deep microchannel
by a second immiscible liquid introduced through identical but symmetrically opposed
microchannels (Objective II). The second involved building a basis for optimal design of
the microfluidic system through determining the effects of various flow parameters and
geometry characteristics on the flow dynamics and topological changes of the multiphase
flow system (Objective III).
The thesis is laid out as follows. Chapter 2 provides a literature review that identifies the
strength and weak points of the different CFD methods, some research gaps that support
the aim of this thesis, and the technical background of the method used. Chapters 3, 4 and
5 report the work undertaken to achieve Objectives I, II and III, respectively; these
chapters are papers, all of which will be submitted to the Physics of Fluids. The final
chapter draws together the work reported here to make overall conclusions and suggest
future work.
6
Literature Review Chapter 2:
2.1 Microfluidics
A brief history of microfluidics 2.1.1
Microfluidics is defined as the study of the behaviour of fluid flow geometrically confined
to a micro scale with dimensions varying roughly between 100 nm and 100 µm [1, 2]. The
potential for microfluidics to revolutionize the energy, chemical processing and medical
fields has attracted significant interest and research effort over the past few decades. In the
1980s, research on microfluidic devices appeared due to the availability of the
micromechanics technology [41]. With the development of the concept of “Micro Total
Analysis System” (µTAS) [42-46], and subsequently Lab-on-a-Chip (LOC) [47-50],
micro-fabrication techniques are capable of making microchannels with complex structures
from resistant and inactive substrate materials which are appropriate for dealing with
chemicals and biological samples [43-45, 51, 52].
Characteristically, a microfluidic system is composed of microfluidic channels, micro-
reservoirs and reaction chambers at micro-meter length scales. It is possible to accomplish
the unit operations including mixing, separation, reaction, detection and analysis, through
one integrated micro-device [53, 54]. There have been several attempts for integration and
parallelization of microfluidic devices. Best examples include Quake et al [48] for
biological systems, Wei et al [55] and Torii et al [56] on droplet generation, and Tetradis-
Meris et al [57] for emulsion formation. Applications of microfluidics have been explored
in colloid science [58], plant biology [59] and process intensifications [60-62]. In
particular, process intensifications involve the micro-chemical engineering technology [63-
67], which can lead to improved quality products, increasing yields, reduction in
investment costs, lower energy consumption, reduced environmental and safety risks [68],
and control of extreme reactions [66, 69-71].
7
Advanced techniques to make microfluidics 2.1.2
Since the beginning of the microfluidic era, the techniques of fabrication of microfluidic
devices have been in constant improvement. Examples of the fabrication techniques
include soft-lithography [72], laser ablation [73], and X-ray lithography [74]. The main
challenges of the common microfabrication techniques are connected with the construction
of open microchannels and assembly of machined substrates to create an integrated
structure. Open microchannels can be made through etching process which is expensive,
and may involve aggressive materials [75].
Among the materials used for fabrication of the microfluidic chips, polydimethylsiloxane
(PDMS) is of particular interest, since it is transparent, flexible, replicable, and low-cost,
however, it swells and deforms in the presence of organic oils. Therefore, more robust
materials such as glass, metal, silicon, and thiolene are needed to be applied for droplet-
based microfluidic systems.
Moreover, significant efforts have to be done to develop functional parts of the
microfluidic system, including microvalves for isolation purposes, and digital pumps for
precise control of flow rates [76]. These difficulties associated with the microfabrication
techniques and materials restrict the experimental research studies. Numerical studies,
therefore, should be called to facilitate building of understanding of flow behaviour in such
systems.
Brief overview of current microfluidic flows 2.1.3
Single phase microfluidic systems are based on continuous liquid flows through
microchannels and are efficiently used for trapping, focusing and/or separation of solid
particles [77], cells [78], and single DNA molecules [79]. Protein analysis through
immobilized microfluidic enzymatic reactors is also another application of the single phase
microfluidics [80]. Single phase microfluidic systems, however, are less suitable for tasks
requiring a high degree of flexibility.
In contrast, multiphase flows provide different mechanisms for improving the performance
of single phase systems. For example, immiscible multiphase microfluidic systems enable
manipulating of discrete volumes of fluids in immiscible phases that benefit large
8
interfacial areas, fast mixing and reduced mass transfer limitations [81]. Examples include
emulsions and droplet-based microfluidics with a wide range of applications from
chemical and material processing to biology and medicine [4].
2.2 Multiphase microfluidics
Fluids of interest in many engineering and industrial applications involve more than a
single phase. Multiphase microfluidic systems play a significant role in drug discovery,
biomedical purposes and many other industrial fluid dynamical problems involving fluid
transport and interfacial phenomena [43]. For instance, biomedical examples in Lab on
Chip devices include emulsions of bacterial cells (solid-liquid-liquid flow) and analysis of
whole blood samples (solid-liquid flow) [49, 82]. Other applications of multiphase micro-
flow systems are synthesis of a small-amount of fine chemicals or active pharmaceutical
ingredients in chemical and pharmaceutical industries, where solid catalysts are used for
gas or liquid reactants [44, 83, 84] . In this case, the hydrodynamics of multiphase flow has
the most important effect on the reactor performance and consequently in the design of a
reactor [44]. Some investigation has been done on the design of microchannel mixers in
order to enhance the advection phenomena in liquid-liquid mixing processes, such as the
micro-mixer designed by Bringer et al [85] for dilute solutions in aqueous drops dispersed
in an oily continuous phase. Liquid-Liquid phase interactions have also been observed in
mono-dispersed emulsions such as water-oil two phase flows in microfluidic devices [11,
57].
2.3 Immiscible Liquid-Liquid Microfluidics
Of particular interest in this project is multiphase flow of immiscible liquids in
microfluidics. Through interaction of immiscible-liquid micro-flows, the flow and
capillary instabilities emerge and threads, jets or monodispersed droplets form. This is the
physical basis for producing emulsions and dispersion of one liquid phase in the other one
[86, 87]. Such microfluidic systems are of relevance to emulsifications used in food [8, 9],
cosmetics [7], pharmaceutical [6, 12] and oil industries [10, 11, 88], plus liquid-liquid
extraction used in drug discovery [5, 89-91]. Figure 2-1 displays a schematic illustration of
9
(a) integrated microfluidic liquid-liquid extraction [5, 92] and (b) water-in-oil-in-water
(W/O/W) emulsion [93].
(a) (b)
Figure 2-1 Examples of liquid-liquid microfluidics; (a) integrated microfluidic liquid-liquid
extraction [92] and (b) water-in-oil-in-water (W/O/W) emulsion in microchannels [93]
Immiscible liquid interfaces are important throughout many areas, particularly chemistry and
biology, because of their unique chemical and interfacial behaviour. There is a capillary
pressure across a curved liquid interface at equilibrium, which is described by Young-
Laplace equation as:
∆𝑃𝑐𝑎𝑝 = 𝛾𝑘 (1)
Where, 𝛾 and 𝑘 are interfacial tension and local mean curvature at the interface, respectively.
According to the equation, the variation of the pressure drop across the interface is a function
of the interface curvature. For instance, the capillary pressure across the interface of a
spherical water droplet with a radius of 1 µm in air is approximately 1000 times, larger than
that of the capillary pressure of the same droplet with a radius of 1 mm [81]. When two
immiscible liquids are brought into contact with each other, various flow patterns and phase
10
distributions can occur due to the interplay between various physical forces such as
interfacial, gravitational, viscous and inertial forces. Figure 2-2 shows the different flow
patterns such as threading, jetting, dripping, tubing, and viscous displacement [14] in the
immiscible-liquid microchannel flow. At micro-scale, because of the laminar nature of the
flow, the flow pattern is dictated by the competition between viscous forces and capillary
force at the interface. The former tends to drag the liquids along the microchannel, while the
latter tends to minimize the interface of the two liquids.
The competition of the various forces is affected by flow parameters, microchannel geometry
and surface chemistry. These effects can be captured by a number of dimensionless numbers
expressing the relative importance of the competing forces [2, 81, 86]. Scaling these forces
based on flow conditions and fluid characteristics helps predict multiphase microfluidic
behaviour, and introduce design guidelines for microfluidic devices.
(a) (b)
Figure 2-2 Possible flow patterns in immiscible-liquid microfluidics at (a) cross-junction [14],
the flow regimes from top to bottom show threading, jetting, dripping, tubing, and viscos
displacement regimes, (b) T-junction [94], the flow patterns from top to bottom show slugs,
monodispersed droplet, droplet populations, parallel flows, parallel flows with wavy interface,
chaotic thin striations flow
One of the most important dimensionless numbers is capillary number (Ca) which is defined
as the relative effect of viscous force to interfacial tension acting across the interface between
two media:
11
𝐶𝑎 =𝜇 𝑈
𝛾 (2)
Where 𝜇, 𝑈 and 𝛾 denote viscosity, velocity and interfacial tension. Capillary number is
calculated for both dispersed phase 𝐶𝑎𝑑 and continuous phase 𝐶𝑎𝑐 . According to literature in
the microfluidic context, Capillary numbers are usually less than 1.0 which means that
capillary forces (or interfacial forces) play a more important role in the flow behaviour than
viscous forces. When Ca is very small around zero, the viscous shearing effect can be
neglected and the interfacial tension is enough to keep the interface away from any
deformation, however by increasing the Ca, deformation of the interface increases and may
lead to formation of droplets and threads. Figure 2-3 shows a typical flow map for an
immiscible-liquid pair [14] based on Ca values of the dispersed and continuous phases. The
flow map reveals how changes in capillary numbers can influence the flow patterns and
instabilities. Five typical flow patterns are indicated by the map including threading, jetting,
dripping, tubing, and displacement. The threading flow pattern refers to a long core thread of
the main channel liquid. The jetting flow pattern refers to a thin thread that emits droplets
smaller than size of the channel at a distance from the junction. In dripping flow regime, the
droplet forms near the junction. The tubing flow pattern is viscous stress controlled and refers
to a viscous core. At low momentum of the continuous phase viscous displacement occurs
[14].
Figure 2-3 Typical Ca-based flow map for immiscible-liquid two-phase microfluidics [14]
12
In addition, the influence of flow rates on the flow pattern and topological changes can be
determined from Reynolds number (Re), which represents the ratio of inertia to viscous
forces:
𝑅𝑒 =𝜌𝑈𝑑
𝜇 (3)
Where ρ and d are fluid density and hydraulic diameter of microchannel, respectively.
Literature shows that the flow rate of about 1 l/s is typically applied in microchannels with
a dimension in the order of 100 m [14, 95]. Using these data, Re can be estimated in the
orders of 0.1 to 1. At a small Reynolds Number (Re < 1), the flow is governed by viscous
stresses and pressure gradients, and inertial effects can be omitted, which reflects the typical
microfluidic characteristics. The other important dimensionless numbers in multiphase
microfluidic systems include Weber number, which implies the influence of inertia with
respect to surface forces (i.e. interfacial tension), and Eotvos number which refers to the ratio
of gravity force to surface forces.
Experimental study of immiscible liquid-liquid microfluidic system 2.3.1
Nakajima et al. [17, 96] and Knight et al. [97] were pioneers for creating monodisperse
emulsions in a microfluidic device, the former through an array of cross-flow microchannel
network and the latter using a hydrodynamic focusing microfluidic system. Similarly,
Thorsen et al [15] employed a T-junction microfluidic configuration for generating dispersed
water droplets in oil emulations, and they demonstrated how the interface instability evolves
due to a competition between shear forces and surface tension. Flow instability usually causes
the formation of monodispersed droplets in microfluidic devices, whereas, a capillary
instability is anticipated to create segmented flows with identical droplets in liquid-liquid
systems [98, 99]. Stone et al. [100] experimentally investigated formation of liquid droplets
in an immiscible liquid through a flow-focusing microfluidic device. They concluded a phase
diagram that illustrates the drop size as a function of flowrate and flowrate ratio of two
liquids. Dummann et al. [101] developed a capillary-microreactor concept and
experimentally studied nitration of a single ring aromatic in an exothermic liquid-liquid
capillary-microreactor which is proposed for intensification of heat and mass transfer.
Dreyfuss et al. [102] used a microfluidic cross-junction for their experiments to examine the
13
role of wetting properties in controlling flow patterns. Moreover, they demonstrated that
phase distribution is influenced by superficial velocities of both continuous and dispersed
phases in immiscible-liquid microchannel flows. Cramer et al. [103] investigated dripping
and jetting flow regimes in co-flowing streams. Furthermore, Okushima et al. [104] reported
a new technique for producing double emulsions (water-in-oil-in-water dispersions) using a
two-step droplet formation in T-shaped microchannels. Over the past decade, different
microchannel configurations including T-junctions [35, 105], Y-junctions [18], cross-
junctions [20, 38], flow focusing [14, 16, 100, 106-109], concentric injection [110] and
membrane emulsification [111-113] have been reported for droplet formation process.
Understanding of flow dynamics and droplet formation mechanisms in the proposed
microfluidic configurations has also been well progressed to facilitate a smart design of
liquid-liquid microfluidic devices. In this context, Garstecki et al. [21] studied the effects of
flow rates and material parameters on the mechanism of droplet formation and breakup in
liquid-liquid emulsions using a microfluidic T-junction. Constraining the liquid-liquid flow
between microchannel walls influences the rupture of an annular liquid core flow into
droplets. At a small Capillary number, less than 10-2
, the shear stress forced on the interface
of the appearing droplet becomes so inadequate that the force is not able to deform
significantly the interface. Thus, the droplet blocks approximately the whole cross-section of
the microchannel and confines the continuous flow to the walls. Garstecki et al. [21]
concluded that increasing Ca in a liquid-liquid system may lead to three different patterns for
the formation of droplets including squeezing, dripping and jetting. Droplet generation in the
squeezing regime is stimulated with pressure instabilities during the breakup process.
A successful example of the microfluidic configuration for producing monodispersed droplets
is microfluidic hydrodynamic focusing wherein a centrally located flow is narrowed or
hydrodynamically focused by two neighbouring flows through cross-like microchannels. For
example, Xu et al. [16] set up a three-stream laminar flow configuration to hydrodynamically
focus a more viscous liquid surrounded by two inviscid liquids to facilitate droplet formation
through control of the flow rate ratio between the continuous phase and the dispersed phase.
The efficiency of droplet formation in microchannels is mainly affected by the nature of
liquids (viscosity and interfacial tension), flow parameters (flowrate ratio, capillary number),
and geometrical characteristics. In this respect, Cubaud et al. [14] performed a
14
comprehensive experimental study on a few Newtonian immiscible liquids in a microfluidic
hydrodynamic focusing device. They investigated the competition between viscosity and
interfacial tension effects on the formation of different flow regimes using hydrodynamic
focusing over a range of viscosities and interfacial tensions at a fixed geometry. Although
their experiments cover a useful range of flowrate ratios, viscosity ratios and Ca, their results
provide an insight to the flow behaviour in the conditions limited to the examined cases. We
meant to develop a reliable simulation to cover a wide range of parameters in order to
increase our predictive capacity for a smart design of immiscible-liquid microfluidic devices.
Despite the various studies on droplet formation and breakup process through different
microfluidic junctions have been reported so far, a detailed study on the effects of
geometrical characteristics and junction/inlet design of microchannels is still missing. The
flow and junction geometry, the cross-section of the channels, the dimensions and the angle
made by the inlet channels all can be influential on flow pattern. There are only a few
researches on the geometry effects reported by literature. For instance, Ménétrier-Deremble
et al. [114] found out the controlling effect of flow geometry in breakup process through
their experiments related to droplet breakup in different-angled microfluidic T-junction
configuration. Also, Tan et al. [115] considered a microfluidic system through a trifurcating
junction to illustrate the effect of junction geometry on droplet coalescence. They deduced
that wider junction lessens the shear force and aligns the droplets to the core section of the
channel and let droplets to create a zigzag pattern resulting in a lateral coalescence. Abate et
al. [116] demonstrated that droplet formation through a flow-focusing device occurs at
higher capillary numbers than to that of a T-shaped microfluidic junction. Furthermore, Priest
et al. [117] designed a Y-shaped microchip for the recovery of copper from leach solutions
in which two liquid phases (organic and aqueous) are contacted in parallel through the main
channel and separated at a Y-junction. They also concluded that the proposed microchip has
greater potential for recycling of volatile liquids. More recently, Li et al. [19] focused on the
droplet formation mechanism in an immiscible liquid-liquid system with T-junction
distributor and noticed that smart geometric design of micro-fluidic devices can affect
controlling the droplet formation on demand. Also, Kunstmann-Olsen et al. [108] studied
experimentally and theoretically a two-dimensional microfluidic cross-flow and showed that
the local geometry of the microchannel-junction affects the hydrodynamic focusing
15
behaviour. We aimed to figure out these effects through our simulation and present a robust
model to predict behaviour of immiscible-liquid flows under the different conditions of
process, material and geometry in a microfluidic hydrodynamic focusing system.
2.4 Numerical study of immiscible liquid-liquid microfluidic system
There is an increasing need to understand and exploit the link between liquid-liquid phase
morphology and fluid dynamics within the phases on the one hand and the processing
conditions that produce them on the other. For example, the droplet size distribution of an
emulsion produced through agitation is a function of the balance between droplet breakup
and coalescence, which can be controlled by the surfactant and stabilizer concentration and
relative velocities of the phases [118-120]. Microfluidic production of encapsulates
provides a further example: in this case, encapsulate morphology can be varied through the
nature of the flow-focusing in the microchannels and, amongst other things, the
continuous-to-dispersed phase flow rate ratio [121, 122]. The challenges faced in
experimentally elucidating these types of relationships are significant, however. For
example, the visualization of the morphologies of the phases and flow fields therein are
still very much in their infancy [123-126]. Models that treat the phases and interfaces
between them explicitly have, therefore, an important role to play in building
understanding of liquid-liquid systems and exploiting this understanding in a systematic
way.
Models of liquid-liquid systems in which the individual phases and interfaces between
them are treated explicitly are long-standing [127-130]. The earliest models, which focus
on droplets in a continuous phase, include those of Taylor [131], Mason and co-workers
[132, 133], Cox and co-workers [134, 135], and Acrivos and co-workers [136-138]
amongst others. Whilst these models were important in building understanding, they are
limited by a good number of the simplifying assumptions [139], including negligible
inertial effects (i.e. small Reynolds numbers), small viscosity ratio ranges, and regular
droplet shapes (e.g. spheres; ellipsoids). Indeed, it was shown by Mason et al. [135] that no
wholly theoretical approach can encompass the entire range of process parameters
adequately. Phenomenological approaches have been adopted to overcome some
limitations associated with wholly analytical models. For example, Maffettone et al. [87]
16
used such an approach, determining its parameters by ensuring it matches analytical model
results in appropriate limits. Such models are, however, also limited by, for example,
underpinning assumptions about droplet shape and the absence of any connection between
the model parameters and underlying fundamentals.
Adoption of wholly numerical approaches can, in principle, overcome limitations faced by
analytical and related models. Some of the earliest numerical models include those of
Acrivos and co-workers [139-141], who studied the deformation and breakup of a viscous
droplet freely suspended in another liquid within extensional and shear flow fields. More
recent examples include the works of Loewenberg, Hinch and Davis [142-148], which are
based on a boundary-integral approach. Afterwards, interface tracking techniques such as
Volume of Fluid (VOF) [130, 149, 150] were used for simulation of immiscible fluids. For
instance, Richards et al. [151] simulated formation of immiscible liquid jets and their
transient to breakup into drops numerically based on the VOF method. Although this
model represent acceptable mass conservation properties, the geometrical characteristics of
the interface, the maintenance of a sharp boundary between the different fluids, and the
surface tension computation are cumbersome to be implemented by VOF, since the volume
fractions are non-smooth functions [149]. As a remedy, a number of techniques such as
Level Set method (LSM) [152, 153] were developed to improve the interface resolution
and include the surface tension computations.
Level Set method set up a smooth distant function in which zero value displays the
position of the interface. As LSM is based on a stream-function formulation and needs to
use a vector potential and solve three elliptic equations, therefore, it becomes more
expensive in three dimensions [154]. Unlike the Level Set method, improved VOF
methods usually involve an efficient interface reconstruction step based on the local
volume fraction and orientation of its gradient. However, it is worthwhile adding that
three-dimensional reconstruction process and as discussed before tracking of curved
interfaces by this method is truly problematic [155]. Alternatively, some techniques made
use of markers to follow the interfaces. Marker methods set up a secondary moving grid
aimed to present precisely the interface location and curvature. Particle-in-Cell
methods[156] which pack the area close to the interface with markers moving with the
flow are another example of this category. The downside of these methods, however, is
17
that they can become rather expensive as the number of markers increases. In addition,
they may suffer from difficulties in controlling the addition or deletion of markers when
the interface is extended or condensed by the flow.
As all these methods involve the use of computational meshes, mesh adjustment is
required as the phases deform. Tracking the interface with substantial deformation and
topological changes is the main challenge of the traditional methods which need high
resolution meshes, mesh refinement, proper mesh aspect ratios and intricate discretization
techniques. Generally, in traditional mesh-based methods, the finite element and finite
volume methods solve the Navier-Stokes equations by integrating the equations over a
mesh of finite elements or volumes. To make the set of nonlinear equations linear,
unknowns are substituted with finite difference equations, and subsequently the linear
equations are solved through an iterative method. To solve the governing equations, the
flow domain has to be properly discretised into tiny volumes or elements resulting in the
solution of a partial differential equation. This process is referred to as mesh or grid
generation. If a proper grid or mesh is not generated, geometric reliability problems will
occur regarding the representation of the computational domain. Furthermore, if a proper
grid is not produced, instabilities will happen when trying to solve the governing
equations. Figure 2-4 illustrates this difficulty with meshing and re-meshing process in
different microfluidic systems. Once a solution has been formed, the flow parameters can
be extracted for each individual cell or element for analysis. These various operations are
complex, computationally expensive and prone to numerical instability [27]. Treatment of
coalescence and rupture of interfaces between phases is extremely challenging if not
impossible in many of these techniques [157]. In conclusion, simulation of immiscible
liquid-liquid systems using mesh-based Computational Fluid Dynamics (CFD) methods
will be computationally very expensive and a formidable task to undertake [27].
Above and beyond the mentioned strengths and weaknesses, modelling of multiphase
micro-flows is particularly challenging because it is not possible to simply adopt a multi-
fluid modelling approach– which treats the immiscible liquids as a homogeneous mixture
without any interfaces –as is typically done for macro-scale counterparts [24, 158]. At
micro-scale, the flow is predominantly laminar, the discrete phases are controllable and of
the order of the size of the geometry. Existence of movable and complex boundaries at the
18
liquid-liquid interface, and liquid-solid contact line dynamics makes the simulation to be
problematic. In such scales, the surface to volume ratio is extremely high and notable,
constitutive relations of liquids are mainly under the influence of the boundary, and the no-
slip boundary condition is not valid. To overcome these issues, we must explicitly model
the behaviour of the individual phases and their interaction at the interface. Explicit
Numerical Simulation (ENS) of multiphase systems takes advantages of investigating
complex phase geometries and topologies and considering multiple time and length scales
in one simulation [26]. As a result, traditional mesh-based CFD methods cannot be
efficiently applied for simulation of multiphase microfluidics. Thus, for some cases such as
immiscible-liquid flows in microchannels with complicated geometries, a more powerful
approach should be called to remove restrictions on mesh entanglement, complex
geometry, Reynolds number and so forth. This study aims to develop a meshless method
for modelling of such systems.
(a) (b)
Figure 2-4 Computational mesh close to the interface for a rising droplet [159] and (b)
adaptive mesh refinement for a droplet impact on a Liquid-Liquid interface [160]
19
Particle-based approaches have been used as a remedy to the issues faced with traditional
mesh-based approaches. Historically, particle methods present the most primitive efforts of
developing meshless numerical schemes. Particle methods have been demonstrated to be
powerful in numerically treatment of large deformation processes and discrete phenomena
[27]. These methods take the advantage of having no inherent restrictions normally on the
permissible deformation of the considered domain in opposition to mesh-based methods.
As for the former, the connectivity between particles is created as a part of computation
and can vary with time which is not the case for 3D meshes. This positive side of particle
methods is very important for substantial deformation simulations as mesh-based methods
may necessitate constant re-meshing to keep well-behaved elements up. Moreover, the
meshless particle method is capable to be connected with a CAD database more
effortlessly than mesh-based approaches, due to the fact that it is not essential to generate
an element grid. The other considerable aspect of particle methods is that the accuracy and
damage of the components can be simply controlled and meshless discretization supplies
precise depiction of geometric object [27, 31].
Lattice Boltzmann method (LBM) [32, 89] is a mesh-based particle method which was
developed for fluid flow applications. The idea behind this method is to construct
simplified kinetic models integrating the essential physics of microscopic or mesoscopic
systems to facilitate the macroscopic averaged parameters follow the desired macroscopic
equations [32]. LBM has been widely employed in numerical simulation of multiphase
microfluidic systems. For instance, Wu et al. [38] presented a lattice Boltzmann model for
simulation of immiscible two-phase flows in a cross-junction microchannel and also
studied the effect of Ca on droplet formation in such a cross-flow systems. Liu et al. [20]
studied the effect of flowrate ratio and capillary number on droplet formation in a
microfluidic cross-flow configuration using a multiphase lattice Boltzmann model. They
used a limited range of capillary numbers (Ca<0.01) and viscosity ratio (𝛌≤1/50), but a
variety of flowrate ratios in their simulation and recognized three different flow regimes
for their different flowrate ratios at a fixed capillary number. Gupta et al. [161] used LBM
to show how the length of the plugs in the squeezing regime are influenced by the width of
the two channels and the depth of the assembly in a microfluidic T-junction. Additionally,
they examined the channel width ratio effect on the droplet formation process in a T-
20
junction microchannel at high capillary numbers. The benefit of LBM, against the
conventional interface tracking methods, lies in its capability that maintains sharp
interfaces without weighty effort. However, this method as well as its less widely-used
antecedent, lattice-gas automata (LGA) [162, 163], are also not without their problems and
limitations. For example, Galilean invariance is lost in LGA, leading to a spurious density-
dependent factor multiplying the inertial term in the momentum conservation equation
whose effect cannot be removed [164]. On the other hand, LBM-based simulation of
immiscible multiphase fluid flows tend to suffer from stability problems [40]. In the LBM,
the spatial space is discretised in a way that it is consistent with the kinetic equation.
Therefore, it requires not only grid, but also the grid has to be uniform. This actually
causes problems at curved boundaries and so development of irregular grids is needed
[27]. In addition, LBM needs the input thermophysical properties to be adjusted by the
microscopic parameters as a result of microscopic dynamics of distribution functions, and
also due to the kinetic nature of the LB method, hydrodynamic boundary conditions are
difficult to satisfy on a grid point exactly [157]. Therefore, specific techniques are required
to enforce boundary conditions. This work is to develop a robust numerical model for
immiscible liquid microfluidic systems that eliminates the aforementioned needs exist for
LBM simulations of microfluidic systems.
An alternative approach is Dissipative Particle Dynamics (DPD) [164-166], which has
been used to study the rheology of complex fluids [167]. DPD is an off-lattice mesoscopic
simulation technique which involves a set of particles moving in continuous space and
each particle represents a cluster of atoms or molecules, and, as a result of the internal
degrees of freedom associated with each particle, the particle-particle interactions include
random and dissipative contributions [168]. The drawback of DPD is determination of
liquid properties and transport coefficients of the system. Also, because of limitation of
small time steps, DPD is not computationally efficient [169, 170].
One preferable particle method that has a much longer history is Smoothed Particle
Hydrodynamics (SPH) [33, 171-173]. In addition to avoiding all the issues associated with
mesh-based methods, SPH has the advantages that it is derived directly from the Navier-
Stokes equations and, hence, can be easily extended to any non-Newtonian matter. It also
has an innate capacity to adapt through time to better capture steep gradients as they form.
21
Explicit numerical simulation of liquid-liquid micro-flows using SPH can be the most
appropriate approach for capturing liquid-liquid interface deformations [26] thanks to its
advantages over the traditional methods. For example, moving and deformable interfaces
can be easily traced using SPH apart from the complexity of the motion of the particles,
and so numerical investigation of multiphase flows through microfluidic devices with
similar behaviour can be reliably and efficiently relied on the Smoothed Particle
Hydrodynamics Method.
Our findings on the subject of multiphase microfluidic systems are still shallow and need
more attempts, however, experimental studies of such micro-scale flows are quite
challenging and expensive, and the analytical solutions are limited to a few simple cases.
Therefore, development of a powerful numerical technique is highly needed for
comprehensive and effective study of multiphase microfluidic flows with the intention of
achieving a better understanding of dynamics of such flows. The significance of this thesis
lies in the development of a new incompressible SPH-based model to the ENS of
immiscible-liquid microchannel flows with particular regard to correctly capture the
liquid-liquid interface. This work contributes in building of a basis for optimal design of
the microfluidic systems through determining the effects of various flow parameters and
geometry characteristics on the dynamics of the multiphase flow system.
2.5 Smoothed Particle Hydrodynamics (SPH) Method
Smoothed Particle Hydrodynamics, considered as the oldest up to date meshless particle
method, was initially developed by Gingold et al. [174] to solve astrophysical problems in
three dimensional open spaces. They introduced SPH as a technique for finding
approximate numerical solutions of the momentum equations by replacing the fluid with a
set of particles [173]. In this technique, an integral representation of a function, called
kernel approximation, is used to solve the governing partial differential equations using a
Lagrangian approach. A weakly compressible SPH algorithm [175] was developed for
simulation of fluids for which the assumption of slightly compressible is valid such as
coastal applications[176]. The weakly compressible SPH method was also used in some
multiphase flows as dust gas flows [177], interfacial flows [178] and so forth.
22
Morris et al. [179] incorporated the standard SPH formulation with a quasi-incompressible
equation of state to model incompressible flows of low Reynolds number. Later on, Shao
et al. [180] presented the incompressible SPH model to study numerically both Newtonian
and non-Newtonian flows based on the concept of projection SPH [175] and the moving
particle semi-implicit method [181]. In this method, an invariant density equation was used
to calculate the required fluid pressures. Moreover, Hu et al. [182, 183] extended the
incompressible SPH model for simulation of multiphase flows under a wide range of
density ratios. In this approach, the standard SPH formulations were modified to account
for the discontinuity of densities across the interface boundary. Correspondingly, Hu et al.
[29] examined the applicability of SPH for simulation of multiphase flows both at macro-
and meso-scales.
In addition to being used to model multiphase fluid systems via the two-fluid model [177,
184], SPH has been used extensively to model such systems in which the interfaces are
explicitly resolved. Many of these ignore surface tension (e.g. [185-190]) and are, hence,
of limited use for liquid-liquid systems. Others have, however, sought to include surface
tension effects. One broad approach is that first proposed by Morris [191], which
embedded the continuum surface force (CSF) approach [192] within SPH. This approach
was disadvantaged by the need to evaluate interface curvature, a numerically challenging
task that is prone to numerical instability. Adams and co-workers [29] addressed this issue,
although computational expense is still an issue, and the approach has since been extended
to systems involving solid surfaces of desired wettability [29, 193] and, strictly
incompressible systems [182, 183, 194, 195]. A second broad approach sees SPH
combined with the Cahn-Hilliard model [196, 197]; this diffuse interface approach is yet to
see wide use. The third and final broad approach to including surface tension within SPH
is via incorporation of inter-particle forces that lead to phase separation. The earliest
example of this approach was due to Nugent and Posch [198] whose inter-particle force
was inspired by the intermolecular interaction term that arises in the van der Waals
equation of state that was used to close their compressible flow SPH model; this approach
has been widely used since (e.g. [199, 200]) as it is far simpler and cheaper than the CSF-
based approach, although it requires calibration through modelling of the Young-Laplace
problem. A variety of other inter-particle force models have also been proposed since,
23
some were inspired by DPD [170, 201], others are density-dependent [202], whilst various
arbitrary forms have also been used, including trigonometric [203] and inverse square law
[204] forms. In all these cases, however, strict incompressibility has not been imposed; this
has only been done once before [205], using the trigonometric interaction model [203].
Although there is an increasing interest in meshless methods specifically SPH, only a
limited number of their application in porous media and microfluidics have been reported
so far [157]. In this context, Tartakovsky et al. [206] utilized a SPH-based simulation to
investigate behaviour of immiscible and miscible fluid flows in porous media. They also
explored the influence of pore scale heterogeneity and anisotropy on such flows.
Additionally, they applied SPH for modelling of reactive transport and mineral
precipitation in porous media [207]. Successful examples of SPH in other applications
include free surface flows and simple unsteady flows in microfluidics [208, 209]. In the
following, a brief description of general SPH formulation is given.
General SPH Formulation 2.5.1
In SPH formulation the various influences of the Navier-Stokes equation are simulated by
a set of forces that act on an individual particle at a location r, where these forces are given
by scalar quantities that are interpolated at a location r by a weighted sum of contributions
from all neighbouring particles inside a cut-off distance h in the space. This can be
expressed in integral form as follows[33]:
𝐴𝑖 = ∫𝐴(��)
Ω
𝑊(𝑟 − ��, ℎ)𝑑�� (4)
By approximating the integral interpolant by a summation interpolant, the numerical
equivalent to Equation (4) is obtained:
𝐴𝑖 =∑𝐴𝑗𝑉𝑗𝑊(𝑟𝑖𝑗, ℎ)
𝑗
(5)
where j is iterated over all particles, 𝑉𝑗 the volume attributed implicitly to the particle j,
𝑟𝑖𝑗 = 𝑟𝑖 − 𝑟𝑗, where r is the position of a particle, and finally A is the scalar quantity that is
24
being interpolated. The summation is over particles which lie within the radius of a circle
centered at 𝑟𝑖. The following relation between volume, mass and density applies[33].
𝑉 =𝑚
𝜌 (6)
where m is the mass and 𝜌 is the density. Combining this we get the basic formulation of
the SPH interpolation function.
𝐴𝑖 =∑𝐴𝑗𝑗
𝑚𝑗
𝜌𝑗𝑊(𝑟𝑖𝑗, ℎ) (7)
The function 𝑊(𝑟𝑖𝑗, ℎ) is the smoothing kernel, which is a scalar weighted function. The
function uses a position 𝑟 and a smoothing length h. This radius can be seen as a cut-off for
how many particles will be considered in the interpolation. This cut-off radius sets 𝑊 = 0
for |𝑟𝑖𝑗|> ℎ [33].
In SPH, the derivatives of a function can be obtained by using the derivatives of the
smoothing kernel that results in the Basic Gradient Approximation Formula [172]:
∇𝐴𝑖 =∑𝐴𝑗𝑚𝑗
𝜌𝑗𝑗
∇𝑊(𝑟𝑖𝑗, ℎ) (8)
∇2𝐴𝑖 =∑𝐴𝑗𝑚𝑗
𝜌𝑗𝑗
∇2𝑊(𝑟𝑖𝑗, ℎ) (9)
These formulations can produce spurious results, and several corrected formulations have
been developed. One of these is the Difference Gradient Approximation Formula, which
has the advantage that the force vanishes exactly when the pressure is constant.
∇𝐴𝑖 =1
𝜌𝑖∑𝑚𝑗 (𝐴𝑗 − 𝐴𝑖)
𝑗
∇𝑊(𝑟𝑖𝑗, ℎ) (10)
The forces between two particles must observe Newton’s Third Law that for every action
there is an equal and opposite reaction. Pair-wise forces must be equal in size with
opposite sign (𝑓𝑖 = −𝑓𝑗). This means that the differentials in momentum equations that
25
create these forces must be symmetrized. Monaghan developed a symmetrisation, referred
to as the Symmetric Gradient Approximation Formula [33]. It is commonly used for the
pressure gradient.
∇𝐴𝑖 = 𝜌𝑖∑𝑚𝑗 (𝐴𝑖𝜌𝑖2
−𝐴𝑗
𝜌𝑗2)
𝑗
∇𝑊(𝑟𝑖𝑗, ℎ) (11)
An alternative symmetrical formulation is as Equation (12) [34]:
∇𝐴𝑖 =∑𝑚𝑗
𝜌𝑖 (𝐴𝑖 + 𝐴𝑗)
𝑗
∇𝑊(𝑟𝑖𝑗, ℎ) (12)
The basic formulation of the Laplacian has been found to be somewhat unstable under
certain conditions, and there exist a wide range of possible corrections. Shao and Lo
established a correction well suited for the correction of the Laplacian in the viscous force
[180].
∇. (1
𝜌∇𝐴) =∑𝑚𝑗
8
(𝜌𝑖 + 𝜌𝑗)2(𝐴𝑖 − 𝐴𝑗). ∇𝑊(𝑟𝑖𝑗, ℎ)
|rij|2+ η2
(13)
To remain the denominator non-zero, 𝜂 variable is used, which is equal to 0.1h.
Neighbour searching algorithm 2.5.2
In the SPH method, only a limited number of particles are inside the support domain of
dimension kh for the target particle, which are applied in the particle approximations.
These particles in the support domain are usually mentioned to as nearest neighbouring
particles (NNP) for that target particle. The procedure of detecting the nearest particles is
known as nearest neighbouring particle searching (NNPS). Unlike a grid based numerical
method, where the position of neighbour grid-cells are well defined once the grids are
given, the nearest neighbouring particles in the SPH for a given particle can vary with
time. The different NNPS approaches which can be used in SPH implementations include
all-pair search, linked-list search algorithm, and tree search algorithm [33].
26
SPH Algorithm 2.5.3
The SPH algorithm approximates a function and its spatial derivatives through averaging
or summation over neighbouring particles. The special features in the SPH coding are
generally involved under the main loop of time integration process, including the
smoothing function and derivative calculation, particle interaction calculation, smoothing
length evolution, spatial derivative estimation, boundary treatment, etc. A typical
procedure for SPH simulation includes three main steps and some sub-steps, as below.
I. Initialization step: includes the input of the initial configuration of the problem
geometry (dimensions and boundary conditions), discretization information of the
initial geometry of particles, material properties, time step and other simulation
control parameters.
II. Main SPH step: contains the major parts in the SPH simulation, and is implemented
in the time integration loop. The following steps are required to be considered into
the time integration process:
Nearest neighbouring particle searching (NNPS)
Calculating the smoothing function (for the summation density approach) and its
derivatives from the generated information of interaction particle pairs
Calculating the density
Calculating the stress term (viscous force and pressure gradient), internal forces
arising from the particle- particle interactions, and the body forces
Updating particle momentum, density, particle position, velocity, and checking
the momentum conservation
III. Output step: the time step reaches to a prescribed one or at some interval, the
resultant information is saved for later analyses or post-processing.
2.6 Conclusion
Even with the recent developments being made in the field of immiscible-liquid
microfluidics, an in-depth understanding of flow behaviour in micro-geometry is needed to
27
improve the design and optimization of the microfluidic devices. The studies have been
done so far in this context, are still sporadic due to variations in the material used for
fabrication, flow rates, channel dimensions and geometries. Experimental studies at such
small scales are quite challenging and expensive, therefore numerical studies have to be
called to become complementary to the experiments. On the other hand, the numerical
studies involving microchannel flow simulations using mesh-based methods face a variety
of obstacles. As discussed, the most important problem is the difficulties along with mesh
generation which would be a problematic task, time consuming and an expensive process
for cases with complex geometry. Another challenge in mesh-based simulations is to track
the movement of the interfacial boundaries in multiphase fluid flows using explicit
numerical simulation (ENS) method, which causes another source for modelling error and
numerical instability. However, meshless particle methods such as SPH have been the
focus recently to develop the next generation of more efficient computational schemes
designed for more complex problems. However, there isn’t any report in the literature on
the dynamics of immiscible-liquid microfluidics which have been studied based on SPH
technique.
The space of parameters which can influence the flow pattern of two liquids interacting in
immiscible-liquid microfluidics is very extensive and involves the nature of the liquids, the
details of flow injections, geometry of the channels, surface characteristics and so forth.
For instance, droplet formation and breakup processes depend on capillary number,
superficial velocities of the continuous and dispersed phase, flowrate ratio, viscosity ratio,
interfacial tension and geometry of microchannel. Despite various experimental and
numerical studies have been reported on droplet-based microfluidics so far, the detailed
flow physics and process parameters is still missing and much effort is still required to
accurately control interfaces and flow patterns, make use of their interfacial properties and
also to capture all the key parameters for design development of microfluidic devices.
28
Modelling of Immiscible Liquid-Liquid Chapter 3:
Systems by Smoothed Particle Hydrodynamics
H. Elekaei Behjati1, M. Navvab Kashani
1, M. J. Biggs
1,2*
1. School of Chemical Engineering, the University of Adelaide, SA 5005,
Australia.
2. School of Science, Loughborough University, Loughborough, LE11 3TU,
UK.
29
Statement of Authorship
Title of Paper Modelling of Immiscible Liquid-Liquid Systems by Smoothed Particle
Hydrodynamics
Publication Status ☐Published ☐Accepted for Publication
☐Submitted for Publication ☒Publication Style
Publication Details Journal of Computational Mechanics
Author Contributions
By signing the Statement of Authorship, each author certifies that their stated contribution to the
publication is accurate and that permission is granted for the publication to be included in the
candidate’s thesis.
Name of Principal
Author (Candidate) Hamideh Elekaei Behjati
Contribution
Performed modelling and coding for simulation. Processed and
undertook bulk of data interpretation. Prepared manuscript as primary
author.
Signature
Date 23/08/2015
Name of Co-author Moein Navvab Kashani
Contribution Assisted with results analysis, data interpretation and manuscript
preparation.
Signature
Date 23/08/2015
Name of Co-author Mark J. Biggs
Contribution
Supervisor. Advised on experimental design, modelling, and data
interpretation. Advised on manuscript preparation, revised manuscript,
and corresponding author.
Signature Date 23/08/2015
30
Abstract
Immiscible liquid systems are ubiquitous in industry, medicine and nature. Understanding
the relationship between the phase morphologies and fluid dynamics within the phases for
the conditions they experience is critical to understanding in these many occurrences. In
this paper, we detail a Smoothed Particle Hydrodynamics (SPH) model that facilitates
building of this understanding. The model includes surface tension, enforces strict
incompressibility, and allows for arbitrary fluid constitutive models. The validity of the
model is demonstrated by applying it to a range of model problems with known solutions,
including a liquid droplet falling under gravity through another quiescent liquid, Young-
Laplace equation, and droplet deformation and rupture under shear whilst confined.
Keywords: Incompressible Smoothed Particle Hydrodynamics; Newtonian liquids;
Immiscible liquid-liquid microfluidics; Droplet deformation; Drag coefficient
31
3.1 Introduction
Processing of immiscible liquid-liquid dispersions occurs widely in the manufacture of
foods [210], pharmaceuticals [211], cosmetics [212], paints [213], and oil [214]. Examples
of processes include emulsification [213, 215], encapsulation [216, 217], multiphase
reaction systems [218, 219], electrochemical processes [220], bioprocesses [221],
separation processes [222] such as liquid-liquid extraction [223], polymer blending [224],
and oil recovery [225]. Processes involving immiscible liquids are also found beyond
industry including in environmental clean-up [226], artificial oxygen carriers [227], oil
spills [228], and pyroclastic flows [229].
There is an increasing need to understand and exploit the link between liquid-liquid phase
morphology and fluid dynamics within the phases on the one hand and the conditions that
lead to them on the other. For example, the droplet size distribution of an emulsion
produced through agitation is a function of the balance between droplet breakup and
coalescence, which can be controlled by the surfactant and stabilizer concentration and
relative velocities of the phases [118-120]. Microfluidic production of encapsulates
provides a further example: in this case, encapsulate morphology can be varied through the
nature of the flow-focusing in the microchannels and, amongst other things, the
continuous-to-dispersed phase flow rate ratio [121, 122]. The challenges faced in
experimentally elucidating these types of relationships are significant, however. For
example, the visualization of the morphologies of the phases and flow fields therein are
still very much in their infancy [123-126]. Models that treat the phases and interfaces
between them explicitly have, therefore, an important role to play in building
understanding of liquid-liquid systems and exploiting this understanding in a systematic
way.
Models of liquid-liquid systems in which the individual phases and interfaces between
them are treated explicitly are long-standing [127-130]. The earliest models, which focus
on droplets in a continuous phase, include those of Taylor [131, 230], Mason and co-
workers [132, 133], Cox and co-workers [134, 135], and Acrivos and co-workers [136-
138] amongst others. Whilst these models were important in building understanding, they
are limited by a good number of simplifying assumptions [139], including negligible
32
inertial effects (i.e. small Reynolds numbers), small viscosity ratio ranges, and regular
droplet shapes (e.g. spheres; ellipsoids). Phenomenological approaches have been adopted
to overcome some limitations associated with wholly analytical models. For example,
Maffettone and Minale [87] used such an approach, determining the model parameters by
ensuring it matches analytical results in appropriate limits. Such models are, however, also
limited by underpinning assumptions such as, for example, specific droplet shapes and the
absence of any connection between the model parameters and underlying fundamentals.
Adoption of wholly numerical approaches can, in principle, overcome limitations faced by
analytical and related models. Some of the earliest numerical models include those of
Acrivos and co-workers [139-141], who studied the deformation and rupture of a viscous
droplet suspended in another liquid within extensional and shear flow fields. More recent
examples include the works of Loewenberg, Hinch and Davis [142-144, 146-148, 231],
which are based on a boundary-integral approach. Interface tracking techniques such as
Volume of Fluid methods (VOF) [150, 232] and level set methods [152, 153] have been
used even more recently. Whilst these works represent major advances in the field, they
have two significant disadvantages that lead to algorithmic complexity, computational
expense and numerical stability challenges [130, 165]: (1) the need to adjust the mesh as
the phases deform; and (2) the need to track the interfaces between the different phases.
The former can be eliminated by selecting a single set of mesh [233] with very high
resolution enough to correctly capture the interface without adjusting the mesh as the
phases deform but this again would be computationally very expensive. Additionally, the
treatment of coalescence and rupture of the interfaces between the phases is extremely
challenging if not impossible in many of these techniques [234, 235].
Particle-based approaches have more recently been used as a remedy to the issues faced by
mesh-based approaches. The most widely-adopted is the lattice Boltzmann method (LBM)
[32, 236]. This method as well as its less widely-used antecedent, lattice-gas automata
(LGA) [162, 163], are also not without their problems and limitations, however. For
example, Galilean invariance is lost in LGA, leading to a spurious density-dependent
factor multiplying the inertial term in the momentum conservation equation whose effect
cannot be removed [164]. LBM-based simulation of immiscible multiphase fluid flows, on
the other hand, tend to suffer from stability problems [40]. An alternative approach is
33
Dissipative Particle Dynamics (DPD) [164-166], which has been used to study the
rheology of complex fluids [167]. DPD is an off-lattice mesoscopic simulation technique
which involves a set of particles moving in continuous space and each particle represents a
cluster of atoms or molecules, and, as a result of the internal degrees of freedom associated
with each particle, the particle-particle interactions include random and dissipative
contributions[168]. However, the disadvantage of DPD is determination of liquid
properties (e.g. viscosity) and transport coefficients of the system such as self-diffusion
coefficient. Since the validity of the equilibrium properties and analytical expressions is
restricted to the limit of small time steps, DPD is not computationally as efficient as LBM
[169, 170]. A related but computationally far cheaper approach [209] of much longer
standing is Smoothed Particle Hydrodynamics (SPH) [171]. In addition to avoiding all the
issues associated with mesh-based methods, SPH has the advantages that it is derived
directly from the Navier-Stokes equations and, hence, can be easily extended to any non-
Newtonian matter.
In addition to being used to model multiphase fluid systems via the two-fluid model [177,
184], SPH has been used extensively to model such systems in which the interfaces are
explicitly resolved. Many of these ignore surface tension (e.g. [185-190]), making them of
limited value when interest lies in liquid-liquid systems where it plays an important role in
dictating behaviour. Some have, however, sought to include surface tension effects. The
first broad approach is that initially proposed by Morris [191], which embeds the
continuum surface force (CSF) approach [192] within SPH. This approach is
disadvantaged by the need to evaluate the curvature of interfaces, a numerically
challenging task that also leads to numerical instabilities. Adams and co-workers [29]
addressed this issue, although computational complexity and expense is still an issue. The
CSF-based SPH approach has since been extended to systems involving solid surfaces of
desired wettability [29, 193] and strictly incompressible systems [182, 183, 194, 195, 237-
240]. A second broad approach sees SPH combined with the Cahn-Hilliard model [196,
197]; this diffuse interface approach is yet to see wide use, however. The final broad
approach to including surface tension within SPH is via incorporation of inter-particle
forces that lead to phase separation. The earliest example of this approach was due to
Nugent and Posch [198], whose inter-particle force was inspired by the intermolecular
34
interaction term that arises in the van der Waals equation of state used to close their
compressible SPH model. This particular model, which is far simpler and cheaper than the
two alternative approaches mentioned above, has seen some use since (e.g. [199, 241])
along with similar models based on other inter-particle force models, including some
inspired by DPD [170, 201] along with various other more arbitrary forms [202], including
trigonometric [203] and inverse square law [204]. In all these cases, however, strict
incompressibility has not been imposed; In this paper, we report incorporation of surface
tension within strictly incompressible SPH framework by including a Lennard-Jones (LJ)
interaction between particles [242] in immiscibility model. The adoption of the LJ
interaction to bring about surface tension has the advantage that it mirrors the almost
universally used approach to incorporating solid boundaries within SPH, thus opening the
way to unifying the treatment of interfaces in SPH. The paper first provides details of the
SPH model and how it is parametrized to yield the desired surface tension. The new
method is then validated by comparing its predictions to analytical results and experiments
for a number of model problems, including a liquid droplet in a linear shear field and the
same falling freely in a quiescent continuous phase.
3.2 The Method
Governing Equations 3.2.1
Smoothed-particle hydrodynamics builds on the Navier–Stokes equations expressed in the
Lagrangian frame, which for an incompressible, isothermal system are of the form
∇. 𝐯 = 0 (1)
𝜌 𝑑𝐯
𝑑𝑡= ∇. 𝛔 + 𝜌𝐠 + 𝐅𝐼 (2)
where 𝜌, 𝐯 , and 𝛔 are the fluid density, velocity and stress tensor, respectively, and 𝐠 is
the acceleration due to gravity or other body forces. The term 𝐅𝐼 is introduced into the
momentum equation here so as to enable the imposition of immiscibility between the
different liquids as described in more detail below. The form of this force is selected in a
35
way that its sum across the entire multiphase system disappears, ensuring overall
conservation of momentum.
The stress tensor is described here by
𝛔 = −𝑃𝐈 + 𝛕 (3)
where 𝑃 is the pressure, 𝐈 the unit tensor, and 𝛕 the shear stress tensor. The latter for a
Newtonian fluid may be expressed as
𝛕 = −𝜇 [ ∇𝐯 + (∇𝐯)𝑇 ] (4)
where 𝜇 is the dynamic viscosity.
SPH Discretization 3.2.2
In SPH, the fluid is represented by a discrete set of particles of mass, 𝑚𝑖, and viscosity, 𝜇𝑖,
that move with the local fluid velocity, 𝐯𝑖. Here, each particle also carries a colour, 𝑐𝑖, to
indicate which of the immiscible fluids it belongs to. The velocity and other quantities
associated with any particle are interpolated at a position, 𝑟, through a weighted
summation of contributions from all neighbouring particles within a compact support of
size 𝑂(ℎ), as illustrated in Figure 3-1.
Figure 3-1 An illustration of an SPH weighting function with compact support sited on
particle-i (shown in red) that leads to the particle interacting with all other particles-j within
the support of size O(h).
kh
Kernel W(r)
i
rij
j
36
For example, the density of a particle-i is given by [172]
𝜌𝑖 =∑𝑚𝑗
𝑗
𝑊(𝑟𝑖𝑗, ℎ) (5)
where 𝑟𝑖𝑗 is the distance between particles i and j.
The pressure gradient associated with particle-i is given by [172, 207]
𝛻𝑃𝑖 = 𝜌𝑖∑𝑚𝑗 (𝑃𝑗
𝜌𝑗2+𝑃𝑖𝜌𝑖2)
𝑗
𝛻𝑖𝑊𝑖𝑗 (6)
where 𝑃𝑖 is the pressure associated with particle-i.
Finally, the divergence of the shear stress tensor attached to a particle-i is given by [173]
(𝛻. 𝝉)𝑖 = 𝜌𝑖∑𝑚𝑗 (𝛕𝑗
𝜌𝑗2+𝛕𝑖𝜌𝑖2
)
𝑗
. 𝛻𝑖𝑊𝑖𝑗 (7)
The components of the shear stress tensor in this expression, which are derived from
Equation (4), are given by
𝜏𝑖𝛼𝛽 = −(∑
𝑚𝑗𝜇𝑗
𝜌𝑗 𝑣𝑖𝑗𝛽
𝑗
𝜕𝑊𝑖𝑗
𝜕𝑥𝑖𝛼 +∑
𝑚𝑗𝜇𝑗
𝜌𝑗 𝑣𝑖𝑗𝛼
𝑗
𝜕𝑊𝑖𝑗
𝜕𝑥𝑖𝛽) (8)
where 𝐯𝑖𝑗 = 𝐯𝑖 − 𝐯𝑗
We have employed the following quintic spline kernel to ensure the second derivatives of
the smoothing kernel that arise in the viscous SPH model are smooth, thus avoiding
instabilities that are known to arise with lower order kernels [33, 179]
37
𝑊(𝑟∗, ℎ)
=7
478𝜋ℎ2×
{
(3 − 𝑟∗)5 − 6(2 − 𝑟∗)5 + 15(1 − 𝑟∗)5 0 ≤ 𝑟∗ < 1
(3 − 𝑟∗)5 − 6(2 − 𝑟∗)5 1 ≤ 𝑟∗ < 2
(3 − 𝑟∗)5 2 ≤ 𝑟∗ < 3 0 𝑟∗ > 3
(9)
where 𝑟∗ = 𝑟 ℎ⁄ is a dimensionless radius.
Immiscibility Model 3.2.3
The immiscibility between the different fluids is enforced through the force, 𝐹𝐼, in
Equation (2), which is dependent on the colour of the particles. The force is based on the
following functional form between particle pairs-ij that was arrived at through
experimentation
𝜙𝑖𝑗 =
{
𝜀𝑖𝑗 (
𝐿0𝑟)12
𝑖𝑓 𝑐𝑖 ≠ 𝑐𝑗
−𝜀𝑖𝑗 (𝐿0𝑟)6
𝑖𝑓 𝑐𝑖 = 𝑐𝑗
(10)
where, 𝐿0 is a reference length, and 𝜀𝑖𝑗 is a parameter that is linked to the relevant
interfacial tension as outlined further below. The first part of this expression ultimately
leads to a short range repulsive force of equal magnitude but opposite direction between
pairs of different coloured SPH particles (i.e. different liquids), whilst the second part leads
to an somewhat longer ranged attractive force of equal magnitude but opposite direction
operating between pairs of particles of the same colour (i.e. same liquid). Although the first
is clearly necessary to cause phase separation, the less intuitive second force was found to
lead to improved behaviour from the model when included.
The force between particles is obtained from the gradient of the expression
(𝐅𝐼)𝑖𝑗 = −∇ϕ𝑖𝑗 (11)
Bringing together equations (10) and (11), the SPH formulation for the force acting on
particle-𝑖 that brings about phase separation is
38
(𝑭𝐼)𝑖 =∑𝑚𝑗
𝜌𝑗𝜙𝑖𝑗 𝛻𝑖𝑊𝑖𝑗
𝑗 (12)
Solution Algorithm 3.2.4
Combining equations (2), (6), (7), and (12) leads to the following SPH formulation for the
momentum equation:
𝑑𝒗𝑖𝑑𝑡
= −∑𝑚𝑗 (𝑃𝑗
𝜌𝑗2+𝑃𝑖𝜌𝑖2
)
𝑗
𝛻𝑖𝑊𝑖𝑗 +∑𝑚𝑗 (𝝉𝑗
𝜌𝑗2+𝝉𝑖𝜌𝑖2
)
𝑗
. 𝛻𝑖𝑊𝑖𝑗
−1
𝜌𝑖∑
𝑚𝑗
𝜌𝑗(𝑭𝐼)𝑗𝛻𝑖𝑊𝑖𝑗
𝑗
(13)
This is solved using a two-step predictor-corrector scheme based on that proposed by
Cummins and Rudman [175] for single phase strictly incompressible flows and later
extended to multiphase flows without surface tension by Shao and Lo [180]. Firstly, an
initial estimate of the particle velocities, 𝐯∗, is made using only the shear stress,
gravitational and surface force terms in Equation (13) at time step-t (indicated by the
subscript-t; particle indices have been dropped for convenience) [175, 180].
𝑣∗ = 𝑣𝑡 + (1
𝜌𝛻. 𝜏 + 𝑔 +
1
𝜌𝐹𝐼)
𝑡
∆𝑡 (14)
where ∆𝑡 is the time step size. The corresponding initial estimates for the particle positions
are then evaluated using [180]
𝒙∗ = 𝒙𝑡 + 𝒗∗∆𝑡 (15)
Use of these initial particle position estimates in Equation (5) would reveal that the
density𝜌∗, is not fixed as desired. Incompressibility is, thus, enforced by correcting the
initial estimates of the particle velocities using [180]
39
𝒗𝑡+1 = 𝒗∗ + ∆𝒗∗∗ (16)
where the velocity correction is evaluated using the pressure gradient term of the
momentum equation only
∆𝒗∗∗ = −1
𝜌∗𝛻𝑃𝑡+1 ∆𝑡 (17)
To obtain the pressure at the new time, 𝑃𝑡+1 , Equation (16) and (17) are combined and the
divergence is taken to give [175]
𝛻. (𝒗𝑡+1 − 𝒗∗
∆𝑡) = −𝛻. (
1
𝜌∗𝛻𝑃𝑡+1 ) (18)
Imposing the incompressibility condition at the new time step, ∇. 𝐯t+1 = 0, leads to the
Pressure Poisson Equation (PPE) [175]
∇. (1
𝜌∗∇𝑃𝑡+1 ) =
∇. 𝐯∗∆t
(19)
The left hand side is discretized based on Shao’s approximation for the Laplacian in SPH
[180], which is a hybrid of a standard SPH first derivative with a finite difference
computation [175]
∇. (1
𝜌∇𝑃)
𝑖
=∑𝑚𝑗 8
(𝜌𝑖 + 𝜌𝑗)2
(𝑃𝑖 − 𝑃𝑗). 𝐫ij. ∇𝑖𝑊𝑖𝑗
|𝐫𝑖𝑗|2+ 𝜔2
𝑗
(20)
where 𝜔 is a small value (e.g. 0.1ℎ ) to ensure the denominator is always non-zero.
The right hand side of equation (19) is discretised in SPH using
(∇. 𝐯∗)𝑖 = 𝜌𝑖∑𝑚𝑗 (𝐯∗𝑗
𝜌𝑗2 +
𝐯∗𝑖𝜌𝑖2) . ∇𝑖𝑊𝑖𝑗
𝑗
(21)
40
Discretization of the PPE equation leads to a system of linear equations, 𝐀𝐲 = 𝐛, in which
𝐲 is the vector of unknown pressure gradients to be determined, and the matrix 𝐀 is not
necessarily positive definite or symmetric. In the present work, the biconjugate gradient
algorithm [243] was used to solve this set of equations.
Once the velocity at the next time step is determined via use of Equation (16), the new
particle positions are finally obtained using
𝐱𝑡+1 = 𝐱𝑡 +1
2(𝐯𝑡 + 𝐯𝑡+1)∆𝑡 (22)
3.3 Results and Discussion
Young-Laplace and Parameterization of Immiscibility Model 3.3.1
The validity of the new method was first assessed by ensuring that a circular droplet of one
liquid is recovered when sitting in a second, quiescent immiscible liquid of the same
density (𝜂 = 𝜌𝑑 𝜌𝑐⁄ = 1) and viscosity (𝜆 = 𝜇𝑑 𝜇𝑐⁄ = 1), in line with physics. This was
done by observing the change in the shape of a stationary droplet over time using a time
step size of ∆𝑡∗ = ∆𝑡√𝛾 𝜌ℎ3⁄ = 0.1 when the drop is initially a square of size 6ℎ × 6ℎ
(ℎ = 1.25𝐿0, where 𝐿0 = 5 × 10−5𝑚) within a periodic domain of 24ℎ × 24ℎ; the total
number of SPH particles was 𝑁𝑝 = 961. As Figure 3-2 shows for a typical simulation, as
expected, the perimeter of the droplet gradually decreases as it changes from its initial
square shape to a circle. This figure also shows that the pressure difference across the
interface between the two phases, shown in dimensionless form (∆𝑃∗ = ∆𝑃ℎ 𝛾⁄ ), increases
as the droplet transforms in shape, once again in line with the physics where the pressure
difference is counter-balanced by the interfacial tension (in this case, 𝛾 = 0.045 N/m,
which corresponds to 𝜀 = 56.25 N/m2).
41
Figure 3-2 Typical variation through time, 𝒕∗ = 𝒕√𝜸 𝝆𝒉𝟑⁄ , of the circumference of an initially
square neutrally buoyant droplet suspended in a second continuous phase and the associated
change in pressure difference, ∆𝑷∗ = ∆𝑷𝒉 𝜸⁄ , across the interface between the two liquids.
Snapshots of the droplet along the transformation pathway are shown at various points; the
continuous phase is not shown for simplicity.
Figure 3-3 shows the variation of the pressure drop across the interface between the drop
and continuous phase as a function of the inverse of the drop size. The linear behaviour
seen in this figure clearly conforms to the Young-Laplace equation
∆𝑃 =
𝑅 (23)
The relationship between the interaction model parameter in equation (10), , and the
interfacial tension, , can be determined by repeating the simulations that lead to Figure 3-
3 for various values of the former. Doing this leads to Figure 3-4, this shows that the
interaction model parameter varies in a linear fashion with the interfacial tension.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.5
0.6
0.7
0.8
0.9
1
0 200 400 600 800
Axi
s Ti
tle
Cd
rop
let
Axis Title𝑡 𝛾 𝜌ℎ3⁄
𝐶𝑑 𝑝 𝑡𝐶 𝑎
⁄ ∆𝑃∗
𝑡∗
∆𝑃∗
𝐶𝑑 𝑝 𝑡𝐶 𝑎
⁄
42
Figure 3-3 Variation of dimensionless pressure drop across the droplet interface with the
inverse of its dimensionless radius, 𝑹∗ = 𝑹 𝒉⁄ , for 𝜸 = 𝟎. 𝟎𝟏𝟒𝟐 N/m (obtained using 𝜺 = 𝟏𝟔
N/m2).
Figure 3-4 Variation of the interfacial tension, 𝜸, with the interaction model parameter that
induces immiscibility here, 𝜺.
Deformation of a Confined Droplet in a Linear Shear Field 3.3.2
The new method was further validated by considering through time the deformation under
linear shear of an initially circular droplet in a second immiscible liquid of the same
density (𝜂 = 𝜌𝑑 𝜌𝑐⁄ = 1) and viscosity (𝜆 = 𝜇𝑑 𝜇𝑐⁄ = 1), Figure 3-5. The initial droplet of
diameter 𝐷 = 17ℎ (ℎ = 1.2𝐿0, where 𝐿0 = 1 × 10−5 m) was located centrally within a
domain of height 𝐻 = 34ℎ and length 𝐿 = 100ℎ; the total number of SPH particles
was 𝑁𝑝 = 4961. To compensate truncation of the support domain of the kernel at the wall
0
5
10
15
20
25
30
35
40
0 0.1 0.2 0.3 0.4
P
* ×1
04
1/R*
0
0.1
0.2
0.3
0.4
0.5
0.6
0 100 200 300 400 500 600 700
[N
.m-1
]
[ N.m-2 ]𝜀 2⁄
𝛾
⁄
43
boundary, at initial state 3h layers of dummy particles are regularly arranged at the other
side of the wall boundary [110, 111]. Equal but opposite velocities of magnitude 𝑈 were
applied to the top and bottom of the domain with no slip boundary conditions (i.e. the top
and bottom were treated as moving solid surfaces) to yield a linear shear field with shear
rate 𝐺 = 2𝑈 𝐻⁄ . The state of the system was evolved through time using timesteps of size
∆𝑡∗ = 𝐺 ∆𝑡 = 0.001.
Under suitable conditions, the droplet can deform to take on an ellipsoidal shape as
illustrated in Figure 3-5. The character of this droplet may be described in part by the
deformation parameter [244]
𝐷𝑓 =𝑎 − 𝑏
𝑎 + 𝑏 (24)
where 𝑎 and 𝑏 are the lengths of the major and minor axes of the ellipsoid, respectively.
The other key parameter characterizing the droplet behaviour is the angle the major axis of
the ellipsoid subtends to the streamlines of the shear field, 𝜑.
Figure 3-5 Droplet shape and velocity field obtained from a typical SPH simulation
with the key characteristics shown.
Figures 3-6 illustrates the deformation and breakup of a droplet under a confined shear
flow through dimensionless times, 𝑡∗ = 𝐺 𝑡 , at Re = 𝜌𝑅2𝐺 𝜇⁄ = 0.1, λ = 1, D H⁄ = 0.5
and Ca = 𝜇𝑅𝐺 𝛾⁄ = 0.5 , a capillary number greater than the critical value where droplet
breakup is ensured. In comparison, Figure 3-7 indicates the velocity field and shape change
of the same droplet for a Ca value smaller than the critical value.
𝑈
𝑈
𝐻
𝐿
44
Figure 3-6 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟓,
𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫 𝑯⁄ = 𝟎. 𝟓.
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5 6 7 8 9 100
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
𝑡∗=0
𝑡∗=1
𝑡∗=3
𝑡∗=5
𝑡∗=9
𝑡∗=12
45
Figure 3-7 Variation of velocity field and droplet morphology with time for 𝐂𝐚 = 𝟎. 𝟐,
𝐑𝐞 = 𝟎. 𝟏, 𝝀 = 𝟏,and 𝑫 𝑯⁄ = 𝟎. 𝟓.
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
1 2 3 4 5 6 7 8 9 10 110
0.5
1
1.5
2
2.5
3
3.5
4
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 120
0.5
1
1.5
2
2.5
3
3.5
4
𝑡∗=0
𝑡∗=1
𝑡∗=3
𝑡∗=5
𝑡∗=7
𝑡∗=10
46
The velocity fields show changes in x-velocity component along the height of the domain.
This induces a velocity field inside the droplet which reduces the shear stress on the
droplet. As a result, the droplet starts elongation and a rotational motion so that a
circulating flow is set up inside the droplet. At higher Ca, viscous forces are more
important than interfacial forces and therefore, the droplet undergoes a larger deformation
and experiences a breakup, as indicated by figure 3-6.
Figure 3-8 compares the deformation parameter as obtained from the SPH simulations at a
confinement ratio of 𝐷 𝐻⁄ = 0.5 with the experimental results of Sibillo et al. [245] and
Taylor’s model for an unconfined droplet in a shear field [244]
𝐷𝑓 =19𝜆 + 16
16𝜆 + 16 𝐶𝑎 (25)
This figure shows that the deformations predicted by SPH match very well the
experimental results up to the Ca = 0.3 limit probed by Sibillo et al. [245]. The results also
match well those predicted by Taylor’s model up to this Capillary Number. However,
beyond this one can see a significant deviation from Taylor’s model, indicating that
confinement has an effect at higher Capillary Numbers where shear is starting to dominate
over surface tension.
Figure 3-8 Variation of the deformation parameter as defined in Equation (24) with Capillary
Number as predicted by SPH with Np=7701 (solid circles with linear fit shown as a line;
R^2=0.99), SPH with Np=4961 (open circles) Taylor theory as defined by Equation (25) (solid
line) [244], and experimental data (open squares) [245].
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5
Df
Ca
47
Figure 3-9 compares the tilt angle of the ellipsoid (see Figure 3-5) as obtained from the
SPH simulations at a confinement ratio of 𝐷 𝐻⁄ = 0.5 with experimental data and the
value predicted from the Taylor model [244]
𝜑 =𝜋
4−(19𝜆 + 16)(2𝜆 + 3)
80(𝜆 + 1) 𝐶𝑎 (26)
Once again, comparison with experiment is very good. The deviation of the SPH values
from Taylor’s theory is modest but grows with Ca, indicating that the tilt angle is more
sensitive to confinement.
Figure 3-9 Variation of the droplet tilt angle with Capillary Number as predicted by SPH
with Np=4961(open circles), SPH with Np=7701 (solid circles), Taylor theory as defined by
Equation (25) (solid line) [244], and the experimental values of Sibillo [246] (open squares).
Deformation of Free-Falling Droplet 3.3.3
The new method was finally validated by considering the descent of an initially stationary
circular droplet under gravity through a second stationary liquid whose density, 𝜌𝑑 =
1150 𝑘𝑔 𝑚3⁄ differed from that of the continuous phase, 𝜌𝑐, by ∆𝜌 = 𝜌𝑑 − 𝜌𝑐 = 150
kg/m3. The droplet was initially of diameter 𝐷 = 14ℎ (ℎ = 1.2𝐿0, where 𝐿0 = 1 ×
10−4𝑚), and was located within a periodic domain of size 34ℎ × 34ℎ; the total number of
SPH particles was 𝑁𝑝 = 1681. The droplet motion and deformation was followed through
time (∆𝑡∗ = ∆𝑡√𝑔 𝐷⁄ = 0.004) until the terminal velocity was reached and the droplet
0
5
10
15
20
25
30
35
40
45
0 0.1 0.2 0.3 0.4 0.5
φ
Ca
48
shape remained unchanged (𝑡∗ = 45). A range of combinations of Morton Number, Mo =
𝑔∆𝜌𝜇𝑐4 𝜌𝑐
2γ3⁄ , Ohnesorge Number, Oh𝑑 = 𝜇𝑑 √ 𝜌𝑑𝛾𝐷⁄ , and Oh𝑐 = 𝜇𝑐 √ 𝜌𝑐𝛾𝐷⁄ and
Eötvös Number, 𝐸𝑜 = 𝑔 ∆𝜌 𝐷2 γ⁄ were considered.
Figure 3-10 compares the variation with time of the droplet descent speed, 𝑢𝑑∗ = 𝑢𝑑 √𝑔𝐷⁄ ,
obtained here from SPH with that predicted by Han and Tryggvason [247] using a finite
difference front tracking method. This figure shows that the two predictions are
qualitatively very similar. Whilst there are some quantitative differences between the two
predictions, these differences are small about 1.36%. Snapshots of the deforming droplet at
various times indicate how the deformation affects the droplet descent speed. After the
initial rise the droplet descent slows down as it starts deforming, until reaches to a steady
state condition.
Figure 3-10 Variation of the droplet descent speed with time as predicted by SPH (dash line)
and Han and Tryggvason [247] (solid line) for 𝛈 = 𝟏. 𝟏𝟓, 𝛌 = 𝟏, 𝐄𝐨 = 𝟏𝟎, 𝐎𝐡𝒅 = 𝟎. 𝟐𝟒 and
𝐌𝐨 = 𝟎. 𝟎𝟒. Snapshots of the deforming droplet are shown at various times
Figure 3-11 shows the evolution of the droplet descent speed with time for different Eötvös
Numbers at 𝜂 = 1.15, 𝑂ℎ𝑑 = 0.23, 𝑂ℎ𝑜 = 1.25. Although the falling velocity ends to a
steady state, the simulation results reveal some dissimilarity in the droplet velocity trends
for the different Eo. According to the figure at the fixed Ohnesorge numbers, the smaller
Eo reaches to the steady state faster, while for the larger Eo, the droplet experiences an
0
0.05
0.1
0.15
0.2
0.25
0.3
0 10 20 30 40 50
Axi
s Ti
tle
Axis Title𝑡∗
𝑢𝑑∗
49
overshoot in the descent speed that implies larger deformation for the falling droplet.
Figures 3-11 (b) and (c) show the droplet shape changes at the different Eötvös Numbers.
Figure 3-11 (a) variation of droplet descent speed with time as predicted by SPH for 𝜼 =𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟎.𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒 (solid line), and 𝑬𝒐 = 𝟏𝟒𝟒 (dash line). Droplet
deformation for (a) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓, 𝑬𝒐 = 𝟐𝟒, and (c) 𝑶𝒉𝒅 = 𝟎. 𝟐𝟑, 𝑶𝒉𝒐 = 𝟏. 𝟐𝟓,
𝑬𝒐 = 𝟏𝟒𝟒 .
Figure 3-12 illustrates the effect of 𝑂ℎ𝑐 and the corresponding Morton numbers on the
droplet shape and descent speed at the fixed 𝐸𝑜 = 24, and 𝑂ℎ𝑑 = 1.16. According to the
figure, the shape change occurs from spherical to oblate and indented oblate as 𝑂ℎ𝑐 and
Mo decrease, implying the less viscous droplet experiences the more deformation. These
results are in agreement with the shape regime maps developed by Han and Tryggvason.
Figure 3-12 (a) variation of droplet descent speed with time as predicted by SPH for 𝜼 =𝟏. 𝟏𝟓, 𝑶𝒉𝒅 = 𝟏.𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 (solid line), and 𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 (dash line). Droplet
deformation for 𝑶𝒉𝒅 = 𝟏. 𝟏𝟔, 𝑬𝒐 = 𝟐𝟒, (b) 𝑶𝒉𝒄 = 𝟎. 𝟎𝟓 and 𝑴𝒐 = 𝟎. 𝟎𝟎𝟎𝟏𝟓, and (c)
𝑶𝒉𝒄 = 𝟏. 𝟐𝟓 and 𝑴𝒐 = 𝟓𝟖
0
1
2
3
4
5
0 1 2 3 4 5
Axi
s Ti
tle
Axis Title𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (× 𝑟𝑑)
−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)
0
1
2
3
4
5
0 1 2 3 4 5
Axi
s Ti
tle
Axis Title𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (× 𝑟𝑑)
−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)
(a) (b) (c)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50 60
Axi
s T
itle
Axis Title
𝑢𝑑∗
𝑡∗
0
1
2
3
4
5
0 1 2 3 4 5
Axi
s Ti
tle
Axis Title𝑥 − 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟𝑑)
−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)
0
1
2
3
4
5
0 1 2 3 4 5
Axi
s Ti
tle
Axis Title𝑥 −𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟𝑑)
−𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜 (×𝑟 𝑑)
(a) (b) (c)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50 60
Axi
s T
itle
Axis Title𝑡∗
𝑢𝑑∗
50
3.4 Conclusions
In this work, a SPH-based model was developed to study immiscible liquid-liquid systems.
Strict incompressibility was enforced through solving the Pressure Poisson Equation. A
surface tension model was proposed to enforce immiscibility between the different fluids.
The model was validated against different model problems including Young-Laplace
equation, droplet deformation and breakup under confined shear field, and a liquid droplet
falling under gravity through another quiescent liquid. The results suggest that the model is
capable to predict phase morphologies and fluid dynamics of immiscible liquid-liquid
systems.
51
Smoothed Particle Hydrodynamics-based Chapter 4:
Simulations of Immiscible Liquid-Liquid Microfluidics
in a Hydrodynamic Focusing Configuration
H. Elekaei Behjati1, M. Navvab Kashani
1, H. Zhang
1, M.J. Biggs
1,2,*
1. School of Chemical Engineering, the University of Adelaide, SA 5005,
Australia.
2. School of Science, Loughborough University, Loughborough, LE11 3TU,
UK.
52
Statement of Authorship
Title of Paper
Smoothed Particle Hydrodynamics-based Simulations of Immiscible
Liquid-Liquid Microfluidics in a Hydrodynamic Focusing
Configuration
Publication Status ☐Published ☐Accepted for Publication
☐Submitted for Publication ☒Publication Style
Publication Details Journal of Physics of Fluids
Author Contributions
By signing the Statement of Authorship, each author certifies that their stated contribution to the
publication is accurate and that permission is granted for the publication to be included in the
candidate’s thesis.
Name of Principal
Author (Candidate) Hamideh Elekaei Behjati
Contribution
Performed modelling and coding for simulation. Processed and
undertook bulk of data interpretation. Prepared manuscript as primary
author.
Signature
Date 23/08/2015
Name of Co-author Moein Navvab Kashani
Contribution Assisted with results analysis, data interpretation and manuscript
preparation.
Signature
Date 23/08/2015
Name of Co-author Hu Zhang
Contribution Provided advices on simulation.
Signature
Date 23/08/2015
Name of Co-author Mark J. Biggs
Contribution
Supervisor. Advised on experimental design, modelling, and data
interpretation. Advised on manuscript preparation, revised manuscript,
and corresponding author.
Signature Date 23/08/2015
53
Abstract
Smoothed Particle Hydrodynamics (SPH) was applied to the study of hydrodynamic
focusing of a liquid flowing in a deep microchannel by a second, immiscible less-viscous
liquid introduced at right angles through identical but symmetrically opposed
microchannels. To investigate the flow dynamics and topological changes of the two
phases, the phases and the interfaces between them were explicitly modelled. The fluids
were truly incompressible, viscous effects were treated rigorously, and surface tension was
included. The model is able to reproduce the experimental results of Cubaud and Mason
(Phys. Fluids. 20, 053302, 2008), including the flow morphology mapping with the
Capillary Numbers of the two liquids, and the characteristics dimensions of the focused
liquid phase as a function of the liquid flow rate ratio. Going beyond the experimental
results, the model was used to build understanding of the velocity fields.
Keywords: Microfluidics; Hydrodynamic focusing; Smoothed Particle Hydrodynamics;
Immiscible liquids; Surface tension.
54
4.1 Introduction
The potential for microfluidics to revolutionize the energy, chemical processing and
medical fields has attracted significant interest and research effort over the last few
decades [1, 4, 81, 248-253]. Whilst many microfluidic systems involve single phase fluids
only (e.g. lab-on-a-chip analysis of biological fluids [254, 255]), microfluidic processing of
multiphase fluids also offers a wealth of opportunities. For example, adding a second
immiscible fluid phase to single-phase microfluidics facilitates high specific interfacial
area, mixing and mass transfer, leading to increased reaction yields for mass transfer-
limited reactions [81, 248, 256, 257]. Multiphase microfluidics also provides possibilities
to precisely control fluid-fluid interfaces and flow patterns for a wide range of applications
ranging from production of novel materials [81, 258, 259] through to medicine [51, 260,
261]. The production of monodispersed droplets of controlled size in micro-emulsions for
pharmaceutical applications [104, 262] is just one example.
In the absence of electric or magnetic fields, the morphology of the phases in isothermal
two-fluid microfluidic flows is dependent on the balance between the two dominant forces
at play – viscous and surface tension – which can be quantified by the Capillary Number,
the relative flow rates of the two phases, the ratio of their viscosities, and the geometry of
the microfluidic pathways. For example, when one liquid (L1) is ‘focused’ by a second
immiscible liquid (L2) introduced through two side channels as illustrated in Figure 1(a),
the phase morphology can take on one of five characters depending on the Capillary
Numbers of the two streams, 𝐶𝑎𝑖, as indicated in Figure 1(b) [14]: (a) threading in which
L1 is focused into a thin thread by L2; (b) jetting where the thread breaks up due to
instabilities to produce droplets of L1 in L2 down-stream of the flow focusing junction; (c)
dripping in which the droplets of L1 are formed at constant frequency at the junction; (d)
tubing in which L1 occupies much of the outlet cross-section; and (e) viscous displacement
where fingers of L1 penetrate into the side channels. The thickness of the thread in the
threading regime, 𝜀, is related to the ratio of the flow rates [14]
𝜀
𝐻≈ (
𝑄12𝑄2
)1 2⁄
(1)
55
In the dripping regime, the droplet length, 𝑙𝑑, is given by [14]
𝑙𝑑𝐻≈1
2(
𝑄2𝑄1 + 𝑄2
𝐶𝑎2 )−0.17
(2)
In the jetting regime, the diameter of the droplets, 𝑑𝑗, and the thread length, 𝑙𝑗 are given by
[14]
𝑑𝑗
𝐻≈ 3.1 (
𝑄12 𝑄2
)1 2⁄
(3)
𝑙𝑗
𝐻≈
8 𝐶𝑗 𝜇1
𝜋𝐻2𝐶𝑎𝑐𝛾( 𝑄1𝑄22
)1 2⁄
(4)
where 𝐶𝑗 is a constant close to unity and 𝐶𝑎𝑐 is the critical capillary number which was
found to be approximately 0.1 [14]. A range of other configurations beyond that of Figure
4-1 have also been studied over the last two decades, including T-junctions [15, 35, 105],
Y-junctions [18], and concentric injection [110] to reveal a rich phase morphology
behaviour that is dependent on the nature of the flow focusing junction. For example, a
cross-junction like that shown in Figure 4-1 allows formation of symmetrical droplets
whilst only asymmetrical drops are possible with T-junctions [263].
.
(a)
𝐻
𝐻𝐶𝑎1 = 𝜇1𝑈1 𝛾⁄
𝑄1
𝑄2 2⁄
𝑄2 2⁄ , 𝐶𝑎2= 𝜇2𝑈2 𝛾⁄
𝑄1+ 𝑄2
56
(b)
Figure 4-1 (a) schematic of cross-junction flow focusing geometry considered in detail in [14]
and here, where 𝑸𝒊, 𝑪𝒂𝒊, 𝝁𝒊 and 𝑼𝒊 = 𝑸𝒊 𝑯𝟐⁄ are the flow rate, Capillary Number, viscosity
and superficial velocity of liquid Li, respectively, and 𝜸𝟏𝟐 is the interfacial tension for the
liquid pair; (b) phase morphology map based on the Capillary Numbers of the liquid L1
(𝑪𝒂𝟏) and the focusing liquid L2 (𝑪𝒂𝟐): (a) threading; (b) jetting; (c) dripping; (d) tubing;
and (e) viscous displacement. The numbers on the map refer to the conditions that were
considered in the study reported on here.
Though there has been extensive study of the effect of fluid properties and flow geometries
on phase morphology, there are still many gaps in understanding due to the sporadic
sampling of the range of materials, fluids, flow rates, channel dimensions and geometries
[264]. Additionally, understanding of the flow fields in microfluidic devices is even less
well explored because of the challenges faced in measuring these at such small scales.
Numerical studies can assist in filling this gap. For instance, De Menech et al [35] studied
transition regimes in a microfluidic T-junction using a phase field model to show the
importance of viscosity ratio for the droplet formation as well as the effect of confinement
b
0.01
0.1
1
0.0001 0.001 0.01 0.1
Ca
1
Ca2
b
c
e
a
d
be
1
2
3
5
7
8
9
10
11
12
13
141516b
a
cd
e
6
4
57
on the shape of droplet and slugs when the size of the side-channel is less than size of the
continuous phase channel. Similarly, the different flow patterns obtained through droplet
formation process in a T-junction microchannel have been numerically investigated by the
same group [250]. Liu and Zhang [265] studied the effects of flowrates, viscosity and
wettability on the process of droplet formation at a microfluidic junction using an
improved phase field lattice Boltzmann method. Also, Gupta and Kumar [161] applied
lattice Boltzmann method (LBM) to show how the dimensions of the channels in a
microfluidic T-junction influence the droplet size. Recently, Ngo et al. [266] investigated
numerically the droplet formation process in a double T-shaped microchannels using a
Volume of Fluid (VOF) model. Liu et al. [267] employed Dissipative Particle Dynamics
method to study multiphase flow behaviour in an inverted Y-shaped microchannel and a
microchannel network. Wu et al. [268] developed a multiphase LBM for droplet formation
in a microfluidic cross-junction flow geometry and demonstrated the dependence of
dripping flow pattern on both Capillary and Weber Numbers. Likewise, Liu et al. [20]
reported three different flow regimes for varied flowrate ratios at a fixed Capillary number
using LBM. Their study was, however, limited to small Capillary Numbers (Ca<0.01) and
viscosity ratios (𝛌1/50). Shi et al. [269] performed a comparative study on the dynamics
of the droplet formation in a microfluidic T-junction and a flow-focusing configuration
using LBM. They indicated that a spherical droplet can be formed through a flow-focusing
geometry, whereas T-junction is likely to produce an elliptical droplet.
Numerical modelling of immiscible liquid-liquid microfluidics requires the explicit
modelling of the two phases and the interfaces between them. This is necessary for two
reasons at least. Firstly, as the characteristic dimensions of the individual phases are
typically comparable to that of the confining geometry, homogenisation approaches are
inappropriate. The second reason is the fact that interest often lays in understanding the
detailed morphology of the phases, something that is not accessible in more mean field
models such as the two-fluid model. Although explicit modelling of the individual phases
and the interfaces between them through space and time is necessary, it is particularly
challenging. If mesh-based numerical methods are used, for example, the evolution of the
mesh in space and time requires complex and expensive algorithms that can bring error
propagation and numerical instabilities [130, 165]. These methods are also often unable to
58
handle the coalescence or break-up of phases (e.g. break-up of the thread in jetting to form
the droplets) [241, 270]. Meshless methods such as Smoothed Particle Hydrodynamics
(SPH) [27, 171] and the more expensive [209] DPD [271, 272] avoid these issues, on the
other hand.
In the work reported in this paper, an incompressible SPH-based model [273] is shown to
be able to reproduce the experimental results obtained by Cubaud and Mason [14] for the
flow-focusing geometry shown in Figure 4-1(a). The emphasis of the study was on the
effect of the flowrate ratio and Capillary Numbers. The paper first presents the details of
the SPH model in brief along with the details of the study. Results are then presented for
the range of conditions shown on the Capillary Number map in Figure 4-1(b) and
compared with the experimental data of Cubaud and Mason [14], thereby demonstrating
that the SPH method is suitable for building detailed understanding of the influence of
fluid properties and process conditions on the phase morphology. The flow fields are also
explored, adding understanding to the experimental results.
4.2 Model and Study Details
Model 4.2.1
A microfluidic right angled cross-junction consisting of three inlet microchannels and one
outlet microchannel as shown in Figure 4-1(a) was considered; the depth of the
microchannels was assumed to be much greater than their width, 𝐻, and study was
restricted to a plane away from the top and bottom of the channels (i.e. the simulations
were limited to two spatial dimensions). A Newtonian liquid L1 of viscosity 𝜇1 entered the
main inlet channel at a volumetric flow rate 𝑄1 = 𝑈1𝐻2. Likewise, a second Newtonian
liquid L2 of viscosity 𝜇2 < 𝜇1 was injected symmetrically into the two lateral inlet
channels each at a volumetric flow rate (𝑄2 2⁄ ) = 𝑈2𝐻2. This configuration leads to the
second liquid hydrodynamically focusing the first as outlined in the Introduction to the
report here.
The SPH-based model proposed in our earlier work [273] for simulation of immiscible
liquid-liquid flows was applied here to study the flow configuration outlined immediately
above; full details of the SPH model can be found in Elekaei et al. [273]. A total of 6562
59
SPH particles were initially arranged uniformly within the microchannels with a spacing of
𝐿0 = 1 × 10−5 m and the fluid particles were assigned an initial speed of 𝑈𝑖 = 𝑄𝑖 𝐻2⁄ . The
particles located in the main microchannel were initially designated as liquid L1, and those
in the two transverse channels as liquid L2. No-slip boundary conditions and confinement
of the SPH particles within the microchannel walls was implemented as described by
[273]. Dirichlet velocity boundary conditions were applied at the inlets and outlet of the
microchannels using a similar approach without, of course, the particle confinement.
Whenever a particle of a liquid leaves the outlet, it is fed back into the simulation through
the appropriate inlet, being sure to maintain symmetry of the lateral flows, thus conserving
the number of particles (and mass) in the simulations. The timestep size used was 1 ×
10−7 s, and the kernel size was 1.4 𝐿0.
Study Details 4.2.2
The study reported here focused on the two immiscible Newtonian liquids summarized in
Table 4-1. The channel cross sectional size was also set to 𝐻 = 100 m for all
simulations. The 16 Capillary Number pairs (𝐶𝑎1, 𝐶𝑎2) shown in Figure 4-1(b) were
considered by varying independently the superficial velocity of the two liquids, 𝑈𝑖 . These
conditions were selected to widely probe the capacity of the model to reproduce the map
determined experimentally by Cubaud and Mason [14], including the boundaries between
the different phase morphologies.
Table 4-1: Properties of liquid pair considered
Property/Fluid Glycerol (L1) PDMS Oil
(L2)
Viscosity (𝜇𝑖; Pa·s) 1.214 0.00459
Surface tension (𝛾12; N/m) 0.027
4.3 Results and Discussion
Figure 4-2 shows the simulation result for the Capillary Number doublet of (0.5, 0.5),
which corresponds to the point 1 in Figure 4-1(b). It is clear that the model is able to
reproduce the threading flow expected at this point. As shown in Figure 4-10, the same can
be said for the three other simulations undertaken in this part of the phase morphology
60
map. More quantitatively, as Figure 4-3 shows, the SPH model is able to predict very well
the variation of the thread thickness, 𝜀, obtained experimentally, which is defined by
equation (1); the relative error between the correlation and the model is on average 8%,
which is anticipated the deviations are smaller than the experimental uncertainties. The last
point in this figure at 𝑄1 𝑄2 = 0.24⁄ shows the upper limit for 𝑄1 𝑄2⁄ in the
aforementioned conditions for which a threading flow regime is valid.
Figure 4-2 also shows the velocity field at steady state, something that is difficult to
measure experimentally. As a result of a flowrate ratio of 𝑄1 𝑄2⁄ = 0.1 , the two branches
of the continuous phase suppress the dispersed phase at the cross-junction and makes a
thinning process which leads to the formation of a long thread of 𝐿1.
Figure 4-2 Representative outcome for threading regime at 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐,
which corresponds to point 1 on Figure 4-1(b).
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𝑥 (𝜇𝑚)
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61
Figure 4-3 Dimensionless thickness of threads as a function of flowrate ratio as evaluated
using the SPH model (open circles) and equation (1) (solid line).
Figure 4-4 shows the SPH prediction at a Capillary Number doublet of (0.1, 0.003), which
corresponds to point 2 on Figure 4-1(b). Once again, it is clear that the model is able to
reproduce the dripping flow expected at this point. As shown in Figure 4-11, the same can
be said for the three other simulations undertaken in this part of the phase morphology
map. Figure 4-5 shows, the SPH model is able to predict very well the variation of the
droplet length, 𝑙𝑑, obtained experimentally, which is defined by equation (2); the relative
error between the correlation and the model is on average less than 8%, which is
anticipated that the error is smaller than the experimental uncertainties.
Figure 4-4 also indicates the evolution of the droplet formation process. At higher flowrate
ratios that allows a larger momentum for the dispersed phase, 𝐿1 is broken into droplets.
According to the figure, initially the velocity of the dispersed phase reaches to its
maximum where occupies the width of the main channel until collapses under the effect of
the continuous phase momentum that acts against interfacial tension, and leads to a
thinning pinch-off process. The built-up pressure in the side channels makes acceleration
and drives L1 to squeeze until reaches to a constant superficial velocity of U1+U2. At the
end of pinching process, breakup occurs and interfaces retract as a result of surface
tension.
0
0.1
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0.4
0.5
0 0.05 0.1 0.15 0.2 0.25 0.3
epsi
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62
(a)
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(c)
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(d)
Figure 4-4 Representative outcome of the SPH model for dynamics of the droplet formation
in dripping regime at 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑, which corresponds to point 2 on Figure 4-
1(b). The snapshots belong to (a) 𝐭∗ = 𝟎. 𝟎𝟕 , (b) 𝐭∗ = 𝟎. 𝟐 , (c) 𝐭∗ = 𝟎. 𝟑𝟖 , and (d) 𝐭∗ = 𝟎. 𝟒 ,
where 𝐭∗ = 𝐔𝟏𝐭 𝐇⁄
Figure 4-5 Dimensionless length of droplet as estimated by the SPH model (open circles) and
equation (2) (solid line)
Figure 4-6 shows the SPH prediction at a Capillary Number doublet of (0.3, 0.005), which
corresponds to point 3 on Figure 1(b). As with the two other cases considered thus far, it is
clear that the SPH model is able to reproduce the jetting flow expected at this point. As
shown in Figure S3, the same can be said for the three other simulations undertaken in this
part of the phase morphology map. Figure 4-7(a) shows, the SPH model is able to predict
very well the variation of the droplet length, 𝑑𝑗, obtained experimentally, which is defined
by equation (3); the relative error between the correlation and the model is on average 6%,
𝑥 (𝜇𝑚)
(𝜇𝑚)
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0.5
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1.5
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Axi
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𝑙 𝑑𝐻⁄
𝑄2 (𝑄1+𝑄2⁄ 𝐶𝑎2 )−0.17
64
which is anticipated that the deviation is smaller than the experimental uncertainties.
Similar to the dripping regime, a droplet formation and pinching process occurs in the
jetting regime but away from the junction after forming a very thin thread of the liquid in
the main channel, 𝐿1. Similarly, Figure 4-7(b) indicates the SPH results for variation of the
thread length of the jet, 𝑙𝑗, as defined by equation (4); the relative error between the
correlation and the model is on average 10%.
(a)
(b)
Figure 4-6 Representative outcome of the SPH model for dynamics of the jetting regime at
𝑪𝒂𝟏 = 𝟎.𝟑 and 𝑪𝒂𝟐 = 𝟎. 𝟎𝟎𝟓, which corresponds to point 3 on Figure 4-1(b)). The snapshots
belong to (a) 𝒕∗ = 𝟎. 𝟒𝟐 , and (b) 𝒕∗ = 𝟎. 𝟒𝟒 , where 𝒕∗ = 𝑼𝟏𝒕 𝑯⁄
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300
400
500
600
𝑥 (𝜇𝑚)
(𝜇𝑚)
𝑥 (𝜇𝑚)
(𝜇𝑚)
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300
400
500
600
65
(a) (b)
Figure 4-7 Dimensionless (a) droplet diameter (equation (3)), and (b) thread length (equation
(4)) in dripping regime as evaluated by the SPH model (open circles) and the related
correlations (solid line).
Figure 4-8 shows the SPH prediction at a Capillary Number doublet of (1.0, 0.006), which
corresponds to point 4 on Figure 4-1(b). As with the all the other cases considered thus far,
it is clear that the SPH model is able to reproduce the expected flow pattern (i.e. tubing).
The controlling agent in tubing regime is the viscous stress by which a long viscous core
occupies the most cross section of the main channel as can be observed by the figure.
Figure 4-8 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 = 𝟏. 𝟎
and 𝑪𝒂𝟐 = 𝟎.𝟎𝟎𝟔, which corresponds to point 4 on Figure 4-1(b).
0
0.2
0.4
0.6
0.8
1
1.2
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Axi
s Ti
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Axis Title𝑄1 𝑄2⁄
𝑑 𝑗𝐻⁄
0
2
4
6
8
10
12
14
7E-11 8E-11 9E-11 1E-10 1.1E-10
Ax
is T
itle
Axis Title 𝑄1𝑄2 2⁄ 0.5
𝑙 𝑗𝐻⁄
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600
𝑥 (𝜇𝑚)
(𝜇𝑚)
66
And, finally, Figure 4-9 shows the SPH prediction at a Capillary Number doublet of (0.19,
0.0002), which corresponds to point 16 on Figure 4-1(b). Viscous displacement regime,
also known as fingering, happens at a large q, when the low momentum of L2 is not
enough to drive L1, and so L1 starts to occupy the side channels.
Figure 4-9 Representative outcome of the SPH model for the tubing regime at 𝑪𝒂𝟏 = 𝟎. 𝟏𝟗
and 𝑪𝒂𝟐 = 𝟎.𝟎𝟎𝟎𝟐, which corresponds to point 16 on Figure 4-1(b).
4.4 Conclusion
An incompressible Smoothed Particle Hydrodynamics-based model was applied to study
immiscible-liquid flows through microfluidic hydrodynamic focusing configuration for a
wide range of capillary numbers (0.001 < 𝐶𝑎 ≤ 1.0). For a Newtonian liquid pair with a
large viscosity ratio, the different flow patterns known as threading, dripping, jetting, and
tubing regime were achieved through our simulations under the effects of the capillary
number of each liquid phase, as expected by the experimental results of Cubaud and
Mason (Phys. Fluids. 20, 053302, 2008). The characteristics dimensions of the liquid phase
in the main channel such as the thickness of the thread in the threading regime, the droplet
length in the dripping regime, and the droplet diameter in the jetting regime were validated
by the experimental results. Moreover, dynamics of the focused liquid flow and the droplet
formation evolution were discussed for different flow patterns.
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4.5 Supplementary Information
(a)
(b)
(c)
Figure 4-10 Representative outcome for threading regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖,
which corresponds to point 12 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟒 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟏, which
corresponds to point 11 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟑 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟐𝟓, which
corresponds to point 10 on Figure 4-1(b).
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500
600
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(a)
(b)
(c)
Figure 4-11 Representative outcome for dripping regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟖𝟓 and 𝐂𝐚𝟐 =𝟎. 𝟎𝟎𝟐 which corresponds to point 15 on Figure 4-1(b), (a) 𝐂𝐚𝟏 = 𝟎. 𝟏 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟏,
which corresponds to point 8 on Figure 4-1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟎𝟓 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟔 which
corresponds to point 7 on Figure 4-1(b).
𝑥 (𝜇𝑚)
(𝜇𝑚)
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600
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(a)
(b)
(c)
Figure 4-12 Representative outcome for jetting regime at (a) 𝐂𝐚𝟏 = 𝟎. 𝟏𝟗 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟓
which corresponds to point 15 on Figure 1(b), (a) 𝐂𝐚𝟏 = 𝟎.𝟏𝟕 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖, which
corresponds to point 14 on Figure 1(b), and (a) 𝐂𝐚𝟏 = 𝟎. 𝟐 and 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟖 which
corresponds to point 13 on Figure 1(b).
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600
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70
Effects of Flow Parameters and Geometrical Chapter 5:
Characteristics on the Dynamics of Two Immiscible
Liquids in a Hydrodynamic Focusing Microgeometry
H. Elekaei Behjati1, M. Navvab Kashani
1, H. Zhang
1, M.J. Biggs
1,2,*
1. School of Chemical Engineering, the University of Adelaide, SA 5005,
Australia.
2. School of Science, Loughborough University, Loughborough, LE11 3TU,
UK.
71
Statement of Authorship
Title of Paper
Effects of Flow Parameters and Geometrical Characteristics on the
Dynamics of Two Immiscible Liquids in a Hydrodynamic Focusing
Microgeometry
Publication Status ☐Published ☐Accepted for Publication
☐Submitted for Publication ☒Publication Style
Publication Details Journal of Physics of Fluids
Author Contributions
By signing the Statement of Authorship, each author certifies that their stated contribution to the
publication is accurate and that permission is granted for the publication to be included in the
candidate’s thesis.
Name of Principal
Author (Candidate) Hamideh Elekaei Behjati
Contribution
Performed modelling and coding for simulation. Processed and
undertook bulk of data interpretation. Prepared manuscript as primary
author.
Signature
Date 23/08/2015
Name of Co-author Moein Navvab Kashani
Contribution Assisted with results analysis, data interpretation and manuscript
preparation.
Signature
Date 23/08/2015
Name of Co-author Hu Zhang
Contribution Provided advices on simulation.
Signature
Date 23/08/2015
Name of Co-author Mark J. Biggs
Contribution
Supervisor. Advised on experimental design, modelling, and data
interpretation. Advised on manuscript preparation, revised manuscript,
and corresponding author.
Signature Date 23/08/2015
72
Abstract
Recently we reported a numerical study on immiscible-liquid flows in hydrodynamic
focusing microchannels using Smoothed Particle Hydrodynamics (SPH) technique. The
optimal design of the system, however, needs a detailed understanding of the dynamics of
flows and the capability to predict such behaviours. To respond to this need, we utilized
our SPH-based simulation to predict the behaviour of the multiphase flow system for a
wider range of flow parameters including the flowrate ratio, viscosity ratio and capillary
number of each liquid phase. The flowrate ratio of the continuous phase to the dispersed
phase affects the flow behaviour and causes different flow patterns depending on 𝐶𝑎1
and 𝐶𝑎2 . The droplet length also appears to be influenced by the both flowrate quantity
and flowrate ratio in dripping regimes. In addition, we discuss the viscosity ratio effects on
the dynamics of the two-phase flow. Subsequently, we explore the effects of geometry
characteristics such as channel size, width ratio, and the angle between the main and side
inlet channels, on the flow dynamics in a hydrodynamic focusing configuration. According
to our findings, increasing the side channel width causes longer droplets, and the right-
angled design effects the most efficient focusing behaviour.
Keywords: Microfluidics; Hydrodynamic focusing; Smoothed Particle Hydrodynamics;
Immiscible liquids; Surface tension.
73
5.1 Introduction
The flow structures formed by two-phase flows in microchannels have attracted significant
scientific and industrial attention in recent years. The importance of the micro-systems lies
in the dominance of interfacial forces such as viscosity and surface tension over bulk
forces, which enable controlled flow patterns, such as droplet formation, with precise
control over the size of the droplets. Such controlled behaviour of two immiscible liquids
interacting in an emulsion inside a microchannel is highly needed for a variety of industrial
applications such as pharmaceuticals, food processing, chemical processing, enhanced oil
recovery, and biotechnology [274-277].
Nakajima et al [17, 96] and Knight et al[97] were pioneers in generating monodisperse
micro-droplets, the former through an array of cross-flow microchannels and the latter
using a hydrodynamic focusing microchannel. Over the past decade, a number of research
studies have reported developments being made in the field of droplet formation and
breakup using different microchannel configurations, including T-junctions [15, 35, 105,
278], Y-junctions [18], Cross-junctions [20, 38], Hydrodynamic focusing and flow
focusing [14, 16, 100, 106-109], Concentric injection [110] and Membrane Emulsification
[111-113]. As is addressed by these studies, a potential strategy to increase the degree of
monodispersity during droplet production in the micro-geometries is to draw from the
characteristic features of multi-stream laminar flow systems, using continuous flows. In
laminar flows, the multi-liquid flows through microchannels without any turbulence;
interfacial forces dominate inertia and make it possible to control the flow structure, size of
the formed droplets and threads by altering the relative flowrates of the multi-stream.
Hydrodynamic focusing is a successful example of monodispersed droplet formation
micro-configuration, wherein a centrally located flow is narrowed or hydrodynamically
focused by two neighbouring flows through cross-like microchannels. Xu et al[16]set up a
three-stream laminar flow configuration to hydrodynamically focus a more viscous liquid
surrounded by two inviscid liquids in order to facilitate a final breakup through control of
the flow rate ratio. Cubaud and Mason [14] performed a comprehensive experimental
study on Newtonian immiscible liquids in a hydrodynamic focusing microfluidic device.
They investigated the competition between viscosity and interfacial tension effects on the
74
creation of capillary threads and droplets using hydrodynamic focusing over a large range
of viscosities and interfacial tensions at a fixed geometry, which provides a useful range of
flowrate ratios, viscosity ratios and Ca. In addition, they continued their experiments on
miscible fluids in order to investigate miscible interfacial morphologies and flow patterns
[109]. Recently, Kunstmann-Olsen et al[108] studied experimentally and theoretically a
two-dimensional microfluidic cross-flow and showed that the local geometry of the
microchannel-junction affects the hydrodynamic focusing behaviour.
As demonstrated by Baroud et al. [277], the parameters by which the flow pattern of
liquids interacting in a microchannel can be influenced, are very extensive and variables
include the nature of the liquids, the details of the flow injections, the geometry of the
channels, the surface characteristics and so forth. Despite various studies on droplet
formation through microfluidic devices reported so far, a full understanding of the flow
physics and the effective process parameters is still missing. On the other side,
experimental studies for microchannel flows are quite challenging and expensive, so
analytical solutions at such a small scale are limited to a few simple cases. Therefore,
numerical studies can be complementary to these experiments. To date, there are only a
limited number of numerical investigations in the literature. For instance, Wu et al. [38]
presented an improved immiscible lattice Boltzmann model for simulation of immiscible
two-phase flows in a cross-junction microchannel and also studied the effect of Ca on
droplet formation in such a cross-flow systems. Furthermore, Liu et al. [20] studied the
effect of flowrate ratios and capillary numbers on droplet formation in a microfluidic
cross-flow configuration, using a multiphase lattice Boltzmann model. They used a limited
range of capillary numbers (Ca<0.01) and viscosity ratios (≤1/50), but a variety of
flowrate ratios in their simulation and recognized three different flow regimes for their
different flowrate ratios at a fixed capillary number. In spite of these numerical
simulations, much effort is still required to accurately control interfaces and capture all the
key parameters for the design development of microfluidic devices.
Nevertheless, modelling of multiphase micro-flows is not easy because of some of their
special properties like free surfaces, moving interfaces, largescale differences in each
dimension, and microscale phenomena [279]. Numerical studies involving microchannel
flow simulations using mesh-based methods face a variety of obstacles. The most
75
important problem lies in the difficulties associated with generating the meshes, which
would be a problematic task, time consuming and an expensive process for problems with
complex geometry. Another challenge in mesh-based simulations is to track the movement
of the interfacial boundaries of multiphase fluid systems. This is an expensive operation
which affords another source for modelling error and numerical instability [130, 165].
Meshless methods have been the focus of research over the past few decades, in order to
develop the next generation of more efficient computational schemes designed for more
complex problems. One preferred meshless method is Smoothed Particle Hydrodynamics
(SPH) [171, 182].
Recently, we presented a novel attempt to model the flow behaviour of two immiscible
liquids in a hydrodynamic focusing micro-geometry using an incompressible Smoothed
Particle Hydrodynamics (SPH) method. The dynamics of the two-phase flows at various
capillary numbers were studied through the SPH-based simulation. Moreover, the
simulation results were compared and validated against the existing experimental data and
the comparison showed good agreement between the model prediction and experimental
outcomes [280]. The efficiency of droplet formation in microchannels is mainly affected
by the nature of the liquids and flow parameters such as viscosity, interfacial tension,
flowrate ratio, capillary number and geometrical characteristics. Therefore, in the present
work, we develop our SPH-based simulation to numerically study the effects of flow
parameters and geometry on immiscible-liquid flow behaviour in hydrodynamic focusing
microfluidics. The results are discussed in three sections covering the effects of the
flowrate and flowrate ratio, the effects of the viscosity ratio, and finally the effects of the
inlet channels’ geometry.
5.2 Model Details
A microfluidic right angled cross-junction consisting of three inlet microchannels and one
outlet microchannel, as shown in Figure 5-1, was considered; the depth of the
microchannels was assumed to be much greater than their width, H, and the study was
restricted to a plane away from the top and bottom of the channels (i.e. the simulations
were limited to two spatial dimensions). A Newtonian liquid, L1, of viscosity 𝜇1 entered
the main inlet channel at a volumetric flow rate of 𝑄1 = 𝑈1𝐻2. Likewise, a second
76
Newtonian liquid L2 of viscosity 𝜇2 < 𝜇1 was injected symmetrically into the two lateral
inlet channels each at a volumetric flow rate (𝑄2 2⁄ ) = 𝑈2𝐻2. This configuration leads to
the second liquid hydrodynamically focusing the first as outlined in the Introduction to the
report here.
Figure 5-1 Schematic of cross-junction flow focusing geometry
The SPH-based model proposed in our earlier work [273] for simulation of immiscible
liquid-liquid flows was applied here to investigate the effects of various flow parameters
and geometry characteristics; full details of the SPH model, simulation parameters and
boundary conditions can be found in Elekaei et al. [273, 280].
5.3 Results and Discussion
Effects of flowrate and flowrate ratio 5.3.1
To investigate the influence of flowrate ratio, we keep the size of the microchannels
constant at 𝐻 = 100 𝜇𝑚, and consider a reference liquid pair and constant flowrate of the
main channel phase 𝑄1, whilst changing the flow rate of the continuous phase 𝑄2 and
thus 𝐶𝑎2. Firstly, we study the effect of flowrate ratio 𝑞 =𝑄2
𝑄1 on the flow behaviour of each
liquid phase in a pair, including an 80 volume percentage of Glycerol in water as L1 with
𝐻
𝐻𝐶𝑎1 = 𝜇1𝑈1 𝛾⁄
𝑄1
𝑄2 2⁄
𝑄2 2⁄ , 𝐶𝑎2= 𝜇2𝑈2 𝛾⁄
𝑄1+ 𝑄2
77
𝜇1 = 0.077 Pa. s and PDMS oil as 𝐿2 with 𝜇2 = 8.2 × 10−4 Pa. s plus the interfacial
tension of 𝛾12 = 0.0304 −1 [14] at constant 𝐶𝑎1.
The simulation results show various flow regimes at different flowrate ratios. Figure 5-2
represents the influence of the flowrate ratio on the flow pattern of the aforementioned
liquid pair with a fixed viscosity ratio of µ1
µ2= 94, for which 𝑄1 = 25 𝜇𝑙 𝑚𝑖 ⁄ and
𝐶𝑎1 = 0.1 are fixed, while 𝑄2 and 𝐶𝑎2 are variable with regard to a range of flowrate
ratios at 0.1, 0.5, 1, 2, 3, 5, 10, and 20.
(a) (b)
(c) (d)
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(e) (f)
(g) (h)
Figure 5-2 Effect of flowrate ratio on the flow pattern at (a) q=0.1, (b) q=0.5, (c) q=1, (d) q=2,
(e) q=3, (f) q=5, (g) q=10, and (h) q=20, with a fixed viscosity ratio of 𝛍𝟏
𝛍𝟐= 𝟗𝟒
The flow structure of the liquid pair changes from the displacement pattern in Figure 5-
2(a) and the tubing regime in Figures 5-2(b) and (c), to dripping and jetting regimes as
shown in Figures 5-2(d) to 5-2(f) respectively, and finally the threading regime in Figures
5-2(g) and (h), as the volume fraction of the continuous phase 𝛼 =𝑄2
𝑄1+ 𝑄2 increases.
Moreover, it can be understood from the figures that the flowrate ratio affects the size of
the formed droplets and the thickness of the threads, as well as the thickness of the core
plug flow in the tubing regime. Comparing Figure 5-1(d) and (e) indicates that the droplet
length in the dipping regime decreases when the flowrate ratio 𝑞 =𝑄2
𝑄1 or the continuous
phase volume fraction 𝛼 increases.
In addition to the flowrate ratio, the flow rate quantity itself can influence the flow regime
and associated features at a constant flowrate ratio. To study this effect, we have
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79
performed a number of simulations in several groups for which various flowrates of the
dispersed phase at a fixed flowrate ratio were considered. Figure 5-3 indicates a flow
pattern map for the above liquid pair, based on the capillary numbers of both liquids, 𝐶𝑎1
and 𝐶𝑎2, and the flowrate ratio q. Note that changes in 𝐶𝑎1 and 𝐶𝑎2, here, were caused
simply by the changes in the flowrates, 𝑄1 and 𝑄2.
Figure 5-3 Flow regime map in terms of 𝐂𝐚𝟏, 𝐂𝐚𝟐 and flowrate ratio. Dripping (solid square),
displacement (solid circles), tubing (solid triangle), threading (solid diamond), and jetting
flow regime (solid pentagon).
According to the Figure 5-3, variations of the flowrate quantities even at the same flowrate
ratio can change the flow regime. In addition, the figure shows that at smaller values
of Q1 for the liquid pair, the dripping regime is achievable for a wider range of q and Ca2.
Furthermore, Figure 5-4 shows how the flowrate quantity can affect the length of the
droplets of the aforementioned liquid pair in the dripping flow regime at a constant
flowrate ratio of 2. Increasing the flowrate of the dispersed phase results in a smaller
droplet and more stabilized tail, as is obvious in the Figures 5-4(a) to (c) and measured by
Figure 5-4(d).
80
(a) (b)
(c) (d)
Figure 5-4 Effect of the flowrate quantities on the droplet size for the constant flowrate ratio
of q=2 and at (a) 𝐐𝟏 = 𝟏𝟎 µ𝐥
𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟎𝟖𝟔 , (b) 𝐐𝟏 = 𝟐𝟓
µ𝐥
𝐦𝐢𝐧 , 𝐂𝐚𝟏 =
𝟎. 𝟏, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟐𝟐 and (c) 𝐐𝟏 = 𝟒𝟎 µ𝐥
𝐦𝐢𝐧 , 𝐂𝐚𝟏 = 𝟎. 𝟏𝟔𝟖, 𝐂𝐚𝟐 = 𝟎. 𝟎𝟎𝟑𝟔 , (d) dimensionless
droplet diameter vs. flowrate of 𝐐𝟏
The effect of the viscosity ratio 5.3.2
Structure of two-phase flows in microchannels is controlled by the domination of one of
the competing forces such as inertia, viscous forces and interfacial tension. In addition, the
relative importance of the viscous stress of one phase acting on the other is a factor in
determining the flow pattern. Viscous forces tend to deform droplets and keep the interface
of the two-liquid phase smooth by dissipating the energy of the perturbations at the
interface. To find out the effects, we have implemented three groups of simulations for
Glycerol as the dispersed phase and PDMS oil as the continuous phase. During each group
of simulations 𝜇1 = 1.214 Pa. s and 𝛾12 = 0.027 −1and other parameters such as the
flowrates and flowrate ratio have been kept constant whilst a range of viscosity ratios,
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(µ
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Po
siti
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(µ
m)
Position (µm)𝑥 (𝜇𝑚)
(𝜇𝑚)
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200 300 400 500 600 700 800
Po
siti
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(µ
m)
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(𝜇𝑚)
1
1.2
1.4
1.6
1.8
2
0 10 20 30 40 50 60
Axi
s Ti
tle
Axis Title𝑄1 [µl ]⁄
dH⁄
81
denoted by 𝜆 =𝜇1
𝜇2 have been considered. Figure 5-5 shows the simulation results for the
fixed flowrate ratio of q=10 and the different viscosity ratios as shown in Figure 5-5(a)
λ=1484, Figure 5-5(b) λ=264, and Figure 5-5(c) λ=24. By decreasing the viscosity ratio
and increasing viscosity of the continuous phase, the viscous stress of 𝐿2 on the interface
increases, which results in a change in the flow pattern from a tubing flow regime to a
threading regime.
(a) (b)
(c)
Figure 5-5 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0036, (b) λ=264 and Ca2=0.02, and
(c) λ=24 and Ca2=0.2, on flow pattern at q=10 and Ca1=0.5
Figure 5-6 indicates how the droplet formation can be influenced by the viscosity ratio at
the flowrate ratio of q=8 and 𝐶𝑎1 = 0.1. As can be seen from the figure, at high values of λ
(when 𝜇2 is very small), there must be an increase in the flowrate of the continuous phase
at the same capillary number of the dispersed phase in order to achieve a dripping flow
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Axi
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Axis Title𝑥 (𝜇𝑚)
(𝜇𝑚)
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600
200 300 400 500 600 700 800
Po
siti
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(µ
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Position (µm)𝑥 (𝜇𝑚)
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(µ
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82
regime in the outlet channel. A greater disparity in viscosities of the two phases allows for
a larger deformation and deviation from droplet generation. Furthermore, the figure
indicates that longer droplets are formed at higher viscosity ratios.
(a) (b)
(c)
Figure 5-6 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.00054, (b) λ=264 and Ca2=0.003,
and (c) λ=24 and Ca2=0.033 on flow pattern at q=8 and Ca1=0.1
Figure 5-7 displays the results of another group of simulations when q=4.4 and 𝐶𝑎1 = 0.3.
By increasing 𝜇2 , the viscous shear of the continuous phase acting on the dispersed phase
increases, resulting in a change in the two-phase flow regime from displacement as shown
in figure 5-7(a) to a jetting regime in figure 5-7(b) and then to a threading flow regime in
figure 5-7(c).
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)
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)
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83
(a) (b)
(c)
Figure 5-7 Effect of viscosity ratio of (a) λ=1484 and Ca2=0.0009, (b) λ=264 and Ca2=0.005
and (c) λ=24 and Ca2=0.054, on flow pattern at q=4.4 and Ca1=0.3
Effect of geometry 5.3.3
In this section, we investigate the effects of the width ratio and inlet angle on the two-
phase flow behaviour in the crossed microchannels.
5.3.3.1 Width ratio effect
The width ratio of the main channel 𝑊2 with the side channel 𝑊1 at a constant channel
height H can affect the length of the droplets in a dripping flow regime. In an attempt to
study this effect, we have implemented the simulations at a fixed width 𝑊1 =
100 𝜇𝑚 and various width ratios, denoted as 𝜁 =𝑊2
𝑊1 . Figures 5-8 represent the simulation
results for effects of 𝜁 on the droplet size when 𝑞 = 2 and 𝐶𝑎1 = 0.04. According to the
figure, the wider the side channel width, the longer the droplet length.
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(µ
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(µ
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Position(µm)𝑥 (𝜇𝑚)
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(µ
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Position (µm)𝑥 (𝜇𝑚)
(𝜇𝑚)
84
(a) (b)
(c)
Figure 5-8 Effect of width ratio of (a) 𝛇 = 𝟎. 𝟓 and Ca2=0.0036, (b) 𝛇 =1 and Ca2=0.0009, and
(c) 𝛇 =2 and Ca2=0.00026 on droplet length at 𝐪 = 𝟐 and 𝐂𝐚𝟏 = 𝟎. 𝟎𝟒
Figure 5-9 summarises the simulation results related to the width ratio effect on the droplet
length as a function of the flowrate ratio at 𝐶𝑎1 = 0.04. The figure confirms that there is a
good agreement between the predicted results and the values calculated from the following
correlation that was obtained by Cubaud and Mason [14] for droplet length measurement
in a dripping regime:
𝑙𝑑𝐻≈1
2(
𝑄2𝑄1 + 𝑄2
𝐶𝑎2 )−0.17
(1)
By increasing the width ratio in a dripping flow regime the size of the droplets increases at
every flowrate ratio in this range. In addition, the changing trends at each 𝜁 are quite
similar with approximately equal difference which shows the identical effects of width
ratio on the droplet size against the different flowrate ratios.
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(µ
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Po
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(µ
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Position (µm)𝑥 (𝜇𝑚)
(𝜇𝑚)
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Po
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(µ
m)
Position (µm)𝑥 (𝜇𝑚)
(𝜇𝑚)
85
Figure 5-9 Normalized droplet length in terms of the width ratio 𝛇 and the flowrate ratio q,
SPH simulation (solid diamond) and Equation (1) (open diamond) for𝛇 = 𝟎. 𝟓, SPH
simulation (solid circle) and Equation (1) (open circle) for 𝛇 = 𝟏, and SPH simulation (solid
square) and Equation (1) (open square) for 𝛇 = 𝟐
5.3.3.2 Inlet angle effect
In order to study the effect of the inlet angle, which is made by the main channel and the
side channel in the cross-like configuration, three different angles were investigated and
compared at the fixed flowrate ratio q=2, viscosity ratio λ=94, and width ratio 𝜁 = 1 with a
width of 100 μm. The results obtained by the simulations at 𝜃 = 45°, 90°, 135° are shown
in Figures 5-10 (a), (b) and (c) respectively. As is observed in the figures, increasing the
inlet angle leads to an increase in the droplet length. In addition, the breakup process is
affected by changing the inlet angle from 90⁰. According to the results, the sharp-angled
design delays the droplet breakup, whereas the obtuse-angled design hastens the
separation. In the right-angled design, the maximum flow focusing is obtained, however,
for a 45°or 135°-angled design, the inclined flow direction of the continuous phase
influences the focusing behaviour unfavourably. This effect arises from the decomposition
of the inertial force into a smaller vertical component plus a horizontal component which is
co-current with the dispersed phase in the 45° design and counter-current with that in the
135°design. Nonetheless, according to Figures 5-8 and 5-10, at a constant flowrate ratio,
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
0 2 4 6 8 10 12
Axi
s Ti
tle
Axis Title
𝑙 𝑑
𝑞
86
the influence of the inlet angle is less important as compared with the effect of the width
ratio.
(a) (b)
(c)
Figure 5-10 Effect of an inlet angle of (a) θ=45⁰, (b) θ=90⁰ and (c) θ=135⁰ on the droplet
length at q=2, Ca1=0.1 and Ca2=0.002
5.4 Conclusion
In this study, we investigated numerically the behaviour of immiscible-liquid pairs
interacting through hydrodynamic focusing microchannels, based on the Smoothed Particle
Hydrodynamics (SPH) method. We discussed the various aspects of the system, including
the effects of the capillary numbers of both dispersed and continuous phases, the influence
of important flow parameters and geometrical characteristics. Different flow patterns
including viscous displacement, tubing, jetting, dripping and threading regimes were
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(μ
m)
Position (μm)𝑥 (𝜇𝑚)
(𝜇𝑚)
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(μ
m)
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87
obtained through variations of the flowrate ratio. The results show that the dripping regime
only happens for q≥1, meaning that L2 must be at least equal to or greater than L1 to
achieve a dripping flow regime. In addition, we indicated that the flowrate quantity and
flowrate ratio affect the droplet length in dripping regimes. We studied the effects of the
viscosity ratio on the flow structures and the dynamics of the droplets. As a consequence, a
large value of λ imposes an increase in the flowrate of the continuous phase with the same
capillary number of the dispersed phase to attain a dripping regime in the outlet channel.
Subsequently, we investigated how the geometry of the channel junction influences the
flow pattern and the hydrodynamic focusing. More specifically, we discussed the effects of
the different width ratios as well as the inlet angle of the main and side channels in cross-
like microfluidic configurations. According to our findings, increasing the side channel
width causes longer droplets, and the right-angled design gives the most efficient focusing
behaviour.
88
Conclusion Chapter 6:
This thesis is focused on outlining work that was focused on developing the application of
a meshless explicit numerical simulation method for the study of immiscible liquid-liquid
flows in microfluidic systems, so as to improve understanding of the fundamentals of
flows in such devices and develop a design capacity for immiscible-liquid microfluidics.
This work was undertaken through two related developments. The first one involved
developing a Smoothed Particle Hydrodynamics (SPH) model for explicit numerical
simulation of immiscible liquid systems. The model includes the surface tension, enforces
strict incompressibility, and allows for arbitrary fluid constitutive models. The validity of
the model was demonstrated by applying it to a range of interfacial problems with known
solutions, including a liquid droplet falling under gravity through another quiescent liquid,
Young-Laplace equation, and the deformation and breakup of a droplet under a shear field,
whilst confined.
The second development in this research has been divided into two parts: the first involved
the application of the proposed model for the study of hydrodynamic focusing of a liquid
flowing in a deep microchannel by a second immiscible liquid introduced at right angles
through identical but symmetrically opposed microchannels. The model was shown to be
able to reproduce the experimental results of Cubaud and Mason (Phys. Fluids. 20,
053302, 2008), including the flow morphology mapping with the Capillary Numbers of the
two liquids, and the characteristic dimensions of the focused liquid phase as a function of
the liquid flow rate ratio. Going beyond the experimental results, the model was used to
build understanding of the velocity fields.
The second part involved building a basis for the optimal design of the microfluidic system
by determining the effects of various flow parameters and geometry characteristics on the
flow dynamics and topological changes of the multiphase flow system. The different flow
patterns, including displacement, tubing, jetting, dripping and threading regimes were
identified as a result of changes in the capillary numbers of the interacting liquids. The
effects of flowrate quantity and flowrate ratio on the droplet length in dripping regimes
were also shown. The viscosity ratio effect was studied and it was demonstrated that a high
value of the viscosity ratio imposes an increase in the flowrate of the continuous phase,
89
with the same capillary number of the dispersed phase, to attain a dripping regime in the
outlet channel. Moreover, it was indicated how the geometry of the channel junction
influences the flow pattern and hydrodynamic focusing. In this regard, the effects of the
different width ratios, as well as the inlet angles of the main and side channels, were
investigated in the cross-like microfluidic configuration. We concluded that increasing the
side channel width causes longer droplets, and the right-angled design gives the most
efficient focusing behaviour. The two-dimensional model developed in this thesis is
expected to predict the parameters and variables with less than 10% deviation from the
experimental results: that is reasonably useful for the prediction of the hydrodynamics of a
3D system. These developments will provide great insights for designing microfluidic
devices involving immiscible liquid-liquid flows.
In terms of future work, there is clearly an opportunity to expand on the work concerned
with different microfluidic configurations such as T-shaped junctions. It would also be of
interest to extend the method to study double-emulsion microfluidic systems. In the case of
the numerical simulation work, this could also be extended to a three-dimensional model.
In addition, the model can be developed for fluids beyond those considered here, including
non-isothermal conditions and non-Newtonian fluids. The standard SPH algorithm
explicitly solves the conservation equations that is not efficient for the prediction of the
rheological properties of non-Newtonian fluids with high viscosity and therefore an
implicit SPH method is also required [281]. Before this happens, however, parallelisation
of the code and adoption of more efficient methods for solving the Pressure Poisson
Equation will be necessary.
90
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