an application of the cahn-hilliard approach to smoothed particle hydrodynamics · 2017-05-02 ·...

Research ArticleAn Application of the Cahn-Hilliard Approach toSmoothed Particle Hydrodynamics

M Hirschler M Huber W Saumlckel P Kunz and U Nieken

Institute of Chemical Process Engineering University of Stuttgart Boblinger Straszlige 78 70199 Stuttgart Germany

Correspondence should be addressed to M Hirschler manuelhirschlericvtuni-stuttgartde

Received 29 August 2013 Accepted 26 December 2013 Published 19 February 2014

Academic Editor Jong-Chun Park

Copyright copy 2014 M Hirschler et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The development of a methodology for the simulation of structure forming processes is highly desirable The smoothed particlehydrodynamics (SPH) approach provides a respective framework for modeling the self-structuring of complex geometries In thispaper we describe a diffusion-controlled phase separation process based on the Cahn-Hilliard approach using the SPH methodAs a novelty for SPH method we derive an approximation for a fourth-order derivative and validate it Since boundary conditionsstrongly affect the solution of the phase separationmodel we apply boundary conditions at free surfaces and solid wallsThe resultsare in good agreement with the universal power law of coarsening and physical theory It is possible to combine the presentedmodelwith existing SPH models

1 Introduction

Phase inversion and phase separation play an important rolein the preparation of precipitation membranes Dozens ofexperiments and simulations have been performed to extendempirical correlations and the understanding of the majoraspects of these processes

Phase separation models based on the Cahn-Hilliardequation [1] describe nucleation and the coarsening dynamics[2] very well The three consecutive stages of coarsen-ing are diffusion-controlled viscous-controlled and inertia-controlled In the first stage we use the free energy ofa system Every system tries to minimize its free energythe strategy is to find a function of the free energy thataccurately describes a mixturersquos physical behavior Assumingthe local free energy depends on the local concentrationand the concentration of the immediate environment Cahnand Hilliard developed a free energy function based onfundamental thermodynamics They calculated the chemicalpotential from the variance of the free energy with respect todensity The chemical potential is the driving force in a lineardriving force approach

After the fundamental work of Cahn and Hilliard sev-eral researchers [2ndash10] developed phase inversion models

divided into model B and model H in the classificationof Hohenberg and Halperin [11] Weikard [12] studied thedynamics and the stability of the Cahn-Hilliard equationSaxena and Caneba Yingrui et al and others [13 14]compared these results with experimental data In the early1960s Lifshitz and Slyozov found a universal power law fordiffusion-controlled stage [15] Several numerical methodsas for instance Lattice Boltzmann method [16] Fast FourierTransformation [17] or Finite Difference method [18] areused to solve the Cahn-Hilliard equation The major dis-advantage of all these methods is the increasing numericaleffort detecting interfaces and geometric changes if diffusionis coupled to hydrodynamic transport We can reduce thenumerical effort of remeshing by using ameshfree simulationmethod One promising meshfree simulation method is thesmoothed particle hydrodynamics method (SPH) [19 20]

For the SPH approach two models for phase separationare developed Okuzono [5] first studied phase separationfor a binary polymer mixture His approach is similar tothe Cahn-Hilliard approach but differs in the main aspectthat diffusion of material is not regarded Instead of that hecoupled two Navier-Stokes equations Okuzono distributeddifferent kinds of particles (eg particles of types A andB) and described the separation of the particles by viscous

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 694894 10 pageshttpdxdoiorg1011552014694894

2 Mathematical Problems in Engineering

and surface tension forces For the chemical potential Oku-zono used the free energy of the system of Ginzburg-Landautype approximation with an order parameter Unlike thepresent approach Okuzono avoids calculating the secondderivative of the density explicitly instead he used the firstderivative two times in a row The results of Okuzono are ingood agreement with the power law for coarsening dynamicsThemodelrsquos major disadvantage is that it only describes phaseseparation if a phase distribution is known a priori

Thieulot et al [21] developed another phase separationmodel for a binary mixture It is based on the GENERICframework They derived the chemical potential directlyfrom the entropy of the fluid It takes the same shapeoriginally suggested by Cahn and Hilliard Their model isthermodynamically consistent but the effects of diffusivematerial transport to viscous and inertia forces are neglected

In this paper we develop a phase separation modelthat describes the diffusion-controlled stage using the Cahn-Hilliard equation It can easily be extended to describe allthree stages of coarsening by combination with existing SPHmodels As a prerequisite we introduce an approach tocalculate the fourth-order derivative in SPH

Most models use periodic boundary conditions Butthe effects of boundary conditions on the solution of theCahn-Hilliard equation are not negligible Therefore wediscuss different kinds of boundary conditions Since anextension tomulticomponentmixtures is straightforward weconsider only a binary fluid mixture in diffusion-controlledstage Analog to [9] one can extend the model for viscous-controlled and inertia-controlled stages

In Section 2 we first review the Cahn-Hilliard equationSubsequently we introduce the SPH model and an approxi-mation of the fourth-order derivative After that we discussdifferent kinds of boundary conditions In Section 3 we vali-date the approach considering a diffusion-controlled binarymixture with periodic boundary conditions Afterwards wedemonstrate the influence of different kinds of boundaryconditions on the solution of the phase separationmodel andconclude in Section 4

2 Materials and Methods

Themodel describes phase separation on amesoscaleThere-fore wemake one fundamental assumptionWe assume com-pleted nucleation on microscopic scale for the descriptionwith a continuummethod The initial state of the system willcontain uniformly distributed nuclei in space

First we review a phase separation model based on theCahn-Hilliard equation Next we introduce the SmoothedParticle Hydrodynamics method (SPH) and an approxima-tion for the fourth-order derivative Then we apply the SPHmethod to the phase separation model For simplicity weneglect viscous and inertia effects (ie the momentum bal-ance)That is valid in case of equalmolar volume density andnegligible excess values Finally since boundary conditionshave an important influence on the solution we focus on freesurfaces and solid walls

21 Phase Separation Model Phase decomposition is theresult of phase changes as it occurs in for example immer-sion precipitation condensation and evaporation A changeof temperature or concentration of a mixture can cause phasedecomposition After exceeding the binodal of the mixturea fluid mixture switches from stable (miscible) to metastableor unstable (immiscible) state respectively In an immisciblesystem nucleation occurs spontaneously after exceeding thespinodal The nuclei grow or disappear depending on theirchemical potentials After infinite time the system reachesits equilibrium state with homogeneous chemical potential inspace

A phase separation model describes the dynamics ofan immiscible fluid mixture based on fundamental ther-modynamics The thermodynamics describes the total freeenergy of the system It is the same energy on microscopicmesoscopic and macroscopic scale

In the late 1950s Cahn andHilliard [1] developed amodelbased on statistical thermodynamics and phenomenologicalobservations The model describes binary phase decompo-sition very well They start with a Taylor series expansionof the free energy density around the homogeneous stateTheir basic assumption was that the free energy density notonly depends on the local concentration but also on theconcentration of the immediate environment That meansthat they do not neglect higher order gradients of theconcentration As shown later by Fife [8] terms of higherorder are necessary for a stable solution in Hilbert space

The free energy results from Legendre transformation ofthe internal energy 119880

119865 = 119880 minus 119879119878 (1)

with 119879 and 119878 as temperature and entropy The free energy isalso

119865 = int119881

119891 (119888 nabla119888 nabla2119888 ) 119889119881 (2)

119881 is the volume of the system We assume an isothermal andisobaric systemThus the free energy density119891 only dependson the concentration 119888 and its derivatives As a result of thework of Cahn and Hilliard [1] we use

119891 (119888 nabla2119888) = 119891

0 (119888) +1

2120581(nabla119888)

2 (3)

to describe the free energy density With (3) (2) yields

119865 = int119881

(1198910 (119888) +

1

2120581(nabla119888)

2)119889119881 (4)

The first term is the free energy density of the homogeneousmixture It depends only on the local concentration Cahnand Hilliard supposed using

1198910 (119888) = prod

119894

(120596 (119879 119888119894) 119888119894) + 119896119861119879sum

119894

119888119894ln (119888119894) (5)

119896119861and119879 are Boltzmann constant and temperature 119894 indicates

the component

Mathematical Problems in Engineering 3

Two parts represent the energy 119880 One part is 120596(119879) thatincludes the internal energy contribution The other part is120581 [1] that includes the interface energy contribution Thelatter part of (5) is the entropy One could also use otherfundamental equations for119891

0based on Flory-Huggins theory

[22 23] or PC-SAFT theory [24] for polymer mixtures

22 Smoothed Particle Hydrodynamics Method TheSmoothed Particle Hydrodynamics method (SPH) isa Lagrangian particle-based and meshfree simulationmethod Originally developed for astrophysical problems[19 20] the relevance of SPH increases in engineeringscience [25]

221 Kernel Approximation and Second Derivative In SPHone calculates a quantity 119860(x) by interpolation of weightedquantities 119860(x1015840) in an appreciable space 119881

ℎ The weighting

function 119882 called kernel function approximates a Dirac-Delta function [19]

119860 (x) = int119881ℎ

119860(x1015840)119882(ℎ x minus x1015840) 119889x1015840 (6)

int119881ℎ

119889x1015840 is the vicinity of x ℎ is the smoothing length within our case a constant value of ℎ = 31119871

0and 119871

0is the

initial particle spacing The smoothing length represents thewidth of theDirac-Delta approximation Kernel functions areusually spline functions see [26] for example In this papera quintic spline is used [26]

In discrete formulation of (6) each particle representsone element of fluid A transition to discrete formulationleads to

119860 (x119894) asymp sum

119895

119898119895

120588119895

119860(x119895)119882(ℎ 119909

119894119895) (7)

119909119894119895is the distance between particles 119894 and 119895119898

119895is the mass of

particle 119895 In the following sections we write 119882119894119895instead of

119882(ℎ 119909119894119895)

For example we could calculate the density with (7) and119860 = 120588 and obtain

120588119894= sum

119895

119898119895119882119894119895 (8)

But because of the particle deficiency near boundaries ofthe computational domain we introduce a so-called Shepardkernel The Shepard kernel renormalizes the kernel functionthat int119881ℎ

119882119894119895119889119881 = 1 Therefore we calculate the density with

120588119894= sum

119895

119898119895

119882119894119895

119882119894119895119904

119882119894119895119904

= sum

119895

119881119895119882119894119895

(9)

119881119895is the volume of particle 119895Brookshaw [27] introduced an approximation of a second

derivative

(nabla sdot 120572nablaA)119894 = sum

119895

119898119895

120588119895

(120572119894+ 120572119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (A119894minus A119895) (10)

120572 is a property of the fluid for example the viscosity x119894119895is

the distance vector between particles 119894 and 119895 Equation (10)is a hybrid formulation for the second derivative using finitedifferences and the first derivative of the kernel function Incontrast to other formulations of the second derivative thisformulation proved to be less sensitive to particle disorderwe will use this as basis for the fourth-order derivative

222 Fourth-Order Derivative Due to the sensitivity of thesecond-order and higher order derivatives of the kernelfunction to particle disorder it is not possible to do stablesimulations using the fourth-order derivative of the kernelfunctionTherefore we approximate the fourth-order deriva-tive of a quantity 119860(x) with a biharmonic formulation

nabla4119860 = nabla

2(nabla2119860) (11)

Thismeans that the chemical potential as well as the transportequation depends on Laplacian A similar idea for a secondderivative consisting of two first derivatives was published byWatkins et al [28] We apply (10) to (11) and get

(nabla sdot 120573nabla11986010158401015840)119894= sum

119895

119898119895

120588119895

(120573119894+ 120573119895)

x119894119895

1199092

119894119895


with

11986010158401015840

119894= (nabla sdot 120572nabla119860)119894 = sum

119895

119898119895

120588119895

(120572119894+ 120572119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (A119894minus A119895)

(13)

120572 and 120573 are properties of the fluid Equation (12) is notsensitive to particle disorder and is in good agreement withthe power law for the fourth-order derivativeWe will discussthis in Section 3

We determine the error of (12) 119864119894 analog to Fatehi and

Manzari [29]

119864119894= ⟨nablanablanablanabla119860⟩119894 minus nablanablanablanabla119860 119894 (14)

A comparison with an expanded Taylor series of 119860 about 119860119894

leads to

119864119894= 4nabla119860|119894sum

119895

120596119895

1

119909119894119895

(sum

119895

120596119895e119894119895e119894119895sdot nabla119894119882119894119895) e119894119895sdot nabla119894119882119894119895

minus2nablanabla119860|119894 sum

119895

120596119895

1

119909119894119895

(sum

119895

120596119895e119894119895x119894119895e119894119895sdot nabla119894119882119894119895)

times e119894119895sdot nabla119894119882119894119895

+ sdot sdot sdot minus 119874(sum

119895119896

sdot sdot sdot ) minus nablanablanablanabla119860119894

(15)

e119894119895and 120596

119895are abbreviated as x

119894119895119909119894119895and 119898

119895120588119895 The leading

term is proportional to the first derivativeThismeans that theerror of the fourth-order derivative is at least proportional to


the error of the first derivativeThe leading term in truncationerror is at equal particle spacing

|nabla119860|119894 times (119874(ℎ4(Δ

ℎ)

2120577+2

)) (16)

and at particle disorder

|nabla119860|119894 times (119874(1198892

119894Δ2(Δ

ℎ)

2120577+2

)) (17)

120577 is the boundary smoothness Δ is the particle spacing and119889119894is the displacement of a particle from regular gridEven though the approximation (12) is not exact it

leads to good and reasonable results as it will be shown inSection 3

223 Governing Equations We consider a binary isother-mal incompressible and equimolar fluid mixture If weneglect viscous and inertia effects the governing equationsare

119863120588

119863119905= 0 (18)

119863119888

119863119905= nabla sdot (119872nabla120583) (19)

120588 is the density and 119888 is the concentration of the mixtureWe assume the mobility coefficient 119872to be constant andindependent of the concentration120583 is the chemical potential

Equation (19) is the general Cahn-Hilliard equation fora multicomponent mixture The chemical potential of acomponent 119894 is defined by the variance of the free energy 119865to the concentration 119888

119894

120583119894=120575119865

120575119888119894

(20)

If we formulate (19) in SPH using (12) 120572 = 120581 and 120573 = 119872 itresults in

119863119888119894

119863119905= sum

119895

119898119895

120588119895

(119872119894+119872119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (120583119894minus 120583119895) (21)

with

120583119894=120597119891

120597119888119894

minus nabla sdot120597119891

120597nabla119888119894

= 120596 (1 minus 2119888119894) + 119896119861119879 (ln (119888

119894) minus ln (1 minus 119888

119894)) minus

1

2120581nabla2119888119894

120581nabla2119888119894= sum

119895

119898119895

120588119895

(120581119894+ 120581119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (119888119894minus 119888119895)

(22)

The model in its presented form is limited to equimolar andincompressible fluid mixtures

23 Boundary Conditions In a large system when boundaryeffects are negligible it is valid to use periodic boundaryconditions to describe phase separation But there remainsystems where boundaries have an effect on the phaseseparation process

Free surfaces influence the phase separation of a fluidmixture There is not a contribution of the boundary to thefree energy of the fluid mixture because of the negligiblephase

A contact angle between two fluids and a solid phaseresults from solid wall boundary conditions This wasobserved in experiments Now we consider both kinds ofboundaries

In SPH we use imaginary particles to model free surfacesby mirroring the particles normal to the boundary This isimportant because of particle deficiency to conserve mass

Near the solid wall the influence of the wall free energyon the free energy of the fluid mixture is not negligible Wetake this into account with an extension of the free energy

119865 = int119881

119891 (119888 nabla2119888) 119889119881 + int

Ω

119891119908 (119888) 119889Ω (23)

by adding a wall free energy correction term119891119908(119888) represents

the influence of the free energy density of the wall on thefluids free energy 119889Ω implies integration over the boundaryIn SPH we reformulate the wall free energy correction termin a volume formulation We assume that the wall energy isindependent of the wall thickness and get

intΩ

119891119908 (119888) 119889Ω =

int119881119891119908 (119888) 119889119881

1198710

(24)

In the literature few approximations of the wall free energyare specified We use the approximation also used by Yue andFeng [30] we derive the chemical potential contribution ofthe wall to the fluidrsquos chemical potential 120575120583

119908(119888) It can be

written as

120575120583119908 (119888) = minus

120574 (12059012 1205901119908 1205902119908 119888)

21198882 cos120572 (25)

120590 is the surface tension coefficient and 1 2 and 119908 stand forcomponent 1 component 2 and wall 120574 is a function of thesurface tension and the concentration 120574 strongly depends onmaterial properties but only slightly on concentration 120572 isa static contact angle In the following simulations we willuse 120574 = 05 independent of concentration because it is areasonable choice according to the parameters of the freeenergy density of the fluid (120596 = 1 119896

119861119879 = 021 120581 = 2)

3 Results and Discussion

First we validate the previously developed model by com-paring the time-dependent interfacial area of the systemwith the universal power law for diffusion-controlled phaseseparation Then we present results of the phase separationwith free surface and solid wall boundary conditions anddiscuss these results


31 Validation In the early 1960s Lifshitz and Slyozov founda universal power law for diffusion-controlled coarsening[15]

Ω (119905) prop 119905minus13

(26)

Ω is the area of the interface and 119905 is the timeWe validate our model using a binary mixture (com-

ponent 1 as 119888 = 0 and component 2 as 119888 = 1) in two-dimensional spaceThe computational domain consists of 100times 100 particles equally spaced with Δ119909 = 1 We use periodicboundary conditions The model parameters are119872 = 1 120596 =

1 119896119861119879 = 021 and 120581 = 2 At 119905 = 0119904 the initial concentration

is 1198880= 04 and the uniformly distributed fluctuation is Δ119888

0=

5119890minus4 The time step is Δ119905 = 0001119904Figure 1 shows a time series of distribution of the con-

centration 119905 = 0 represents the initial state and 119905 = 30000

represents the equilibrium state of the system The gray scalerepresents the composition of the mixture We represent theconcentration of component 1 bright and the concentration ofcomponent 2 dark The time increases from left to right andtop down

The fluidmixture decomposes into two phases At 119905 = infinthe system is in equilibrium state The equilibrium shape intwo dimensions is a circle of one phase

Next we track the interface size Ω over time 119905 andcompare it with the power law (26) We analyzed Ω atdifferent times with Cannyrsquos graphical analysis method [31]

Figure 2 shows the interface size Ω over time 119905 Interfacesize and time are plotted in double logarithmic referenceframe Crosses in Figure 2(a) represent simulation results forregular particle distribution and the solid line represents thepower law

Simulation results and the power law within resolutionerror are in good agreement with regard to the gradient Attimes 119905 lt 500 the interface size is under the resolution ofthe computational domain At times 119905 = infin the system isin equilibrium For quantitative comparison of the surfacearea calculated with the SPH method we solved the sameequation on a regular grid with 10 000 grid points discretizedwith finite differences (FD) The error between SPH and FDis defined as

120598 (119905) =(ΩSPH (119905) minus ΩFD (119905))

2

ΩFD(119905)2

(27)

and plotted over time (Figure 2(b)) Within resolution errorthe error between both methods is low

The effect of particle disorder is investigated by repeatingthe previous simulation with particles randomly displacedin space by plusmn5 of the regular position In Figure 2(a)points represent the time-dependent surface area versus timeAgain we found good agreement with the power law andquantitative agreement with regular particle distributionTheerror to FD (see Figure 2(b)) is within the same range as forregular particle distribution

32 Effects of Boundary Conditions In the previous sim-ulation we used periodic boundary conditions Now we

investigate the effect of free surfaces and solid walls on phaseseparation

First we investigate free surface boundary conditionsThe simulation parameters are the previous ones exceptfor the computational domain In this and the followingsimulations we choose the particle number equal to 50 times 50to reduce the computation time We only use free surfaceboundary conditions at all boundaries As described inSection 2 the gradient of the chemical potential is zero at theboundary

n sdot nabla1205831003816100381610038161003816120597Ω = 0 (28)

n is the normal vector of the boundaryThis means that noneof the components preferably accumulates at the boundary

Consider the equilibrium of this system The system is inequilibrium when the interface size between both phases isminimal The shape of the equilibrium depends on the initialcomposition of the fluid mixture There exist two possibleshapes of the equilibrium One shape is in two-dimensionalcase a circle of one phase The other shape is a layering ofboth phases Now we analyze the initial composition of thefluid mixture for both shapes

Consider a circle inside of a square If layering of bothphases is the equilibrium shape the circumference 119880 = 120587 sdot 119889

will be larger than the length of the edge of a square 119910

119880 gt 119910 (29)

119889 is the diameter of the circleThe limiting composition fromthe volume of the circle 119860 = 120587(1198892)

2 If we know the initialcomposition the area of the circle at equilibrium is knownWe get the perimeter of the circle with simple conversions Ifwe assume the limiting case 119880 = 119910 we get two solutions

1198880=

1

4120587= 00796

1 minus1

4120587= 09204

(30)

Consequently the range of the initial composition for layer-ing is 00796 lt 119888

0lt 09204

Figures 3(a)ndash3(f) show simulation results with freeboundary conditions and 119888

0= 04 From (29) we expect

layering of both phases as equilibrium shape At time 119905 = infinthe equilibrium shape actually is a layering (see Figure 3)

For quantitative comparison of these results we alsocalculated the same system using finite differences on aregular grid with 2500 grid points Figures 3(g)ndash3(i) showthe results at three different times It is not exactly thesame because the random initial conditions are differentNevertheless the evolution of phase decomposition is similarand the equilibrium states are identical

Next we qualitatively investigate solid wall boundaryconditions As described in Section 2 we use a wall freeenergy correction term (25) to describe the effect of a wallon a fluid mixture For validation of the wall free energycorrection term we consider a droplet of one fluid phasesurrounded by a second fluid phase on a solid wall Keepin mind that a contact angle is defined by the intersection


(a) (b) (c)

(d) (e) (f)

Figure 1 Time series of the fluid mixture at 119905 = 0 (a) 119905 = 500 (b) 119905 = 1500 (c) 119905 = 3000 (d) 119905 = 6000 (e) and 119905 = infin (f) Results withperiodic boundary conditions and 100 times 100 particles Gray scale shows the composition Dark represents 119888 = 1 Parameters are 119888

0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

104

103

102

104

103

102

Time log(t)

Surfa

ce ar

ealog(Ω)

SPH regularSPH irregular

FDPower law

(a)

100

10minus1

10minus2

10minus3

10minus4

Erro

rlog(t)

104

103

102

Time log(t)


(b)

Figure 2 (a) Interface size Ω over time 119905 in double logarithmic reference frame Crosses and dots represent simulation results of SPH withregular and irregular particle distribution The dashed line shows simulation results with finite differences The solid line represents thegradient of the power law (26) (b) Error 120598 between the simulation results of finite differences and SPH for regular and irregular particledistributions over time 119905


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 3 Time series of composition of the fluid mixture at 119905 = 0 (a) 119905 = 15 (b) 119905 = 50 (c) 119905 = 150 (d) 119905 = 500 (e) and 119905 = infin (f) Resultswith SPH and free surfaces boundary conditions and 50 times 50 particles Gray scale shows the composition Dark represents 119888 = 1 Parametersare 1198880= 04 120596 = 1 119896

119861119879 = 021 and 120581 = 2 (g)ndash(i) are results with FD and Neumann boundary conditions and 2500 grid points at 119905 = 15 (g)

119905 = 50 (h) and 119905 = infin (i)

of a fluid-fluid interface and a fluid-solid interface Themagnitude of the contact angle depends on the wall freeenergy correction term We specify an equilibrium contactangle of the droplet and compare it with the equilibriumcontact angle of the droplet in the simulation

The computational domain and the simulation parame-ters except for 119888

0and Δ119888

0 are the previous ones Figure 4

top left shows the initial state of the composition Thecomposition of the smaller square is 119888

0= 099 and of the

larger one 1198880= 001Δ119888

0is zeroWe apply solid wall boundary

conditions to all boundariesWe vary the specified contact angle between 120572 = 0

∘ and120572 = 180

∘ Figure 4 shows the results All specified staticcontact angles are obtainedThe interface between the phasesis smooth This is obvious in cases 120572 = 0

∘ and 120572 = 180∘

In the first case it seems that the contact angle is not stilllarger than zero in the second case it seems that the dropletmoves away from the boundary Regarding the smoothness of


(a) (b) (c)

(d) (e) (f)

Figure 4 Equilibrium state of composition of the fluid mixture with various contact angles Initial state (a) 120572 = 0∘ (b) 120572 = 60

∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180

∘ (f) Results with wall boundary conditions and 50 times 50 particles Parameters are 120596 = 1 119896119861119879 = 021 and 120581 = 2

the interface the droplet is in infinitesimal small contact withthe boundary If we increase the volume of component 2 incase of 120572 = 0

∘ the complete wall is wettedAs a last point we qualitatively consider phase separation

of a binary fluidmixture in a boxThe computational domainand parameters are the previous ones We apply solid wallboundary conditions to all boundaries The contact angleis 120572 = 0

∘ This means that all walls are fully wetted withcomponent 2

Figure 5 shows the results At equilibrium 119905 = infin thesolid wall is fully wetted with component 2Thus if the circleis centered in the box the minimum of the free energy ofthe mixture is obtained For contact angle 120572 = 180

∘ (notshown) we obtain the same equilibrium state with changedcomposition of mixture

4 Conclusions

We developed a SPH approach of a phase separationmodel The model describes diffusion-controlled equimolarcoarsening dynamics based on the Cahn-Hilliard equationThe model is extensible to multicomponent systems andnonequimolar systems as shown in the literature Sincea diffusion equation can be combined with existing SPH

models see for example [32] combination of the presentmodel with existing SPH models is possible

We calculated the chemical potential of a fluid mixturefrom free energy and its second derivative The gradient ofthe chemical potential is proportional to the diffusion fluxsee transport equation (19) In this way the free energy ofimmiscible components intrinsically takes care of surfacetension effects on coarsening dynamics because of thesecond derivative of concentration Since phases with smallervolume have larger chemical potentials because of largercurvature the system is driven from many small volumes toone large volume Therefore larger volumes will grow whilesmaller ones disappear

The novelty of the present SPH model is a direct Cahn-Hilliard approach using a fourth-order derivative of a quan-tity119860 Because of its biharmonic property we split the fourth-order derivative into two derivatives of second-order

We compared the time-dependent interface size of thecomputational domain with the universal power law fordiffusion-controlled phase separation and we found goodagreement with the power law

In the last part we investigated free surface and solid wallboundary conditions We validated the boundary conditionwith the equilibrium state of the system For solidwall bound-ary conditions we introduced a phenomenological approach


(a) (b) (c)

(d) (e) (f)

Figure 5 Time series of composition of the fluid mixture at 119905 = 0 (a) 119905 = 50 (b) 119905 = 200 (c) 119905 = 400 (d) 119905 = 1000 (e) and 119905 = infin (f) Resultswith solid wall boundary conditions and 50 times 50 particles Gray scale shows the composition Dark represents 119888 = 1 Parameters are 119888

0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

This approach is based on a wall free energy correction termWe applied the solid wall boundary conditions to modelstatic contact angles Finally we applied solid wall boundaryconditions to present model

The extension to three or more component systems isstraightforward as seen in the literature [9] Coupling ofthe transport equation to the momentum balance is possiblein straightforward way For nonequimolar fluid mixturesvariable mass or volume of the SPH particles should beconsidered

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The work is supported by the German Science Foundation(DFG) SFB 716 project andwithin the funding programOpenAccess Publishing

References

[1] J W Cahn and J E Hilliard ldquoFree energy of a nonuniformsystemmdashI Interfacial free energyrdquo The Journal of ChemicalPhysics vol 28 no 2 pp 258ndash267 1958

[2] H Tanaka ldquoViscoelastic phase separationrdquo Journal of PhysicsCondensed Matter vol 12 no 15 pp R207ndashR264 2000

[3] R Koningsveld ldquoLiquid-liquid phase separation in multicom-ponent polymer solutionsmdashI Statement of the problem anddescription of methods of calculationrdquo Journal of PolymerScience A vol 6 pp 305ndash323 1968

[4] Y Oono and S Puri ldquoStudy of phase-separation dynamics byuse of cell dynamical systemsmdashIModelingrdquo Physical Review Avol 38 no 1 pp 434ndash453 1988

[5] T Okuzono ldquoSmoothed-particle method for phase separationin polymermixturesrdquoPhysical Review E vol 56 no 4 pp 4416ndash4426 1997

[6] J Lowengrub and L Truskinovsky ldquoQuasi-incompressibleCahn-Hilliard fluids and topological transitionsrdquo Proceedings ofThe Royal Society of London A vol 454 no 1978 pp 2617ndash26541998

[7] D Jacqmin ldquoCalculation of two-phase Navier-Stokes flowsusing phase-field modelingrdquo Journal of Computational Physicsvol 155 no 1 pp 96ndash127 1999


[8] P C Fife ldquoModels for phase separation and their mathematicsrdquoElectronic Journal of Differential Equations vol 48 pp 1ndash262000

[9] B Zhou and A C Powell ldquoPhase field simulations of earlystage structure formation during immersion precipitation ofpolymeric membranes in 2D and 3Drdquo Journal of MembraneScience vol 268 no 2 pp 150ndash164 2006

[10] H Liu T Sengul and S Wang ldquoDynamic transitions forquasilinear systems and Cahn-Hilliard equation with Onsagermobilityrdquo Journal of Mathematical Physics vol 53 no 2 ArticleID 023518 2012

[11] P C Hohenberg and B I Halperin ldquoTheory of dynamic criticalphenomenardquo Reviews of Modern Physics vol 49 no 3 pp 435ndash479 1977

[12] U Weikard Numerische losungen der cahn-hilliard-gleichungund der cahn-larche-gleichung [PhD thesis] RheinischeFriedrich-Wilhelms-Universitat Bonn Germany 2002

[13] R Saxena andG T Caneba ldquoStudies of spinodal decompositionin a ternary polymer-solvent-nonsolvent systemrdquo PolymerEngineering and Science vol 42 no 5 pp 1019ndash1031 2002

[14] S Yingrui WMing D Kazmer C Barry and J Mead ldquoExperi-mental validation of numerical simulation for phase decompo-sition of a binary polymer thin film on a patterned substraterdquo inProceedings of the Society of Plastics Engineers Annual TechnicalConference (ANTEC rsquo07) pp 2916ndash2920 May 2007

[15] I M Lifshitz and V V Slyozov ldquoThe kinetics of precipitationfrom supersaturated solid solutionsrdquo Journal of Physics andChemistry of Solids vol 19 no 1-2 pp 35ndash50 1961

[16] E Orlandini M R Swift and J M Yeomans ldquoA latticeboltzmann model of binary-fluid mixturesrdquo EurophysicsLetters vol 32 no 6 pp 463ndash468 1995

[17] M I M Copetti and C M Elliott ldquoKinetics of phasedecomposition processes Numerical solutions to Cahn-Hilliard equationrdquoMaterials Science and Technology vol 6 no3 pp 273ndash283 1990

[18] B Zhou Simulations of polymeric membrane formation in 2dand 3d [PhD thesis] Massachusetts Institute of Technology2006

[19] R A Gingold and J J Monaghan ldquoSmoothed particlehydrodynamics theory and application to non-spherical starsrdquoMonthly Notices of the Royal Astronomical Society vol 181 pp375ndash389 1977

[20] L B Lucy ldquoA numerical approach to the testing of the fissionhypothesisrdquo The Astronomical Journal vol 82 pp 1013ndash10241977

[21] C Thieulot L P B M Janssen and P Espanol ldquoSmoothedparticle hydrodynamics model for phase separating fluidmixturesmdashI General equationsrdquo Physical Review E vol 72 no1 Article ID 016713 2005

[22] P J Flory ldquoThemodynamics of high polymer solutionsrdquo TheJournal of Chemical Physics vol 10 no 1 pp 51ndash61 1942

[23] M L Huggins ldquoSome properties of solutions of long-chaincompoundsrdquo Journal of Physical Chemistry vol 46 no 1 pp151ndash158 1942

[24] J Gross and G Sadowski ldquoPerturbed-chain SAFT an equationof state based on a perturbation theory for chain moleculesrdquoIndustrial and Engineering Chemistry Research vol 40 no 4pp 1244ndash1260 2001

[25] J J Monaghan ldquoSmoothed particle hydrodynamics and itsdiverse applicationsrdquo Annual Review of Fluid Mechanics vol44 pp 323ndash346 2012

[26] J J Monaghan ldquoSmoothed particle hydrodynamicsrdquo AnnualReview of Astronomy and Astrophysics vol 30 pp 543ndash5741992

[27] L Brookshaw ldquoA method of calculating radiative heat diffusionin particle simulationsrdquo Proceedings of the Astronomical Societyof Australia vol 6 no 2 pp 207ndash210 1985

[28] S J Watkins A S Bhattal N Francis J A Turner and AP Whitworth ldquoA new prescription for viscosity in SmoothedParticle Hydro dynamicsrdquo Astronomy and Astrophysics vol 119no 1 pp 177ndash187 1996

[29] R Fatehi and M T Manzari ldquoError estimation in smoothedparticle hydrodynamics and a new scheme for secondderivativesrdquo Computers amp Mathematics with Applications vol61 no 2 pp 482ndash498 2011

[30] P Yue and J J Feng ldquoWall energy relaxation in the Cahn-Hilliard model for moving contact linesrdquo Physics of Fluids vol23 no 1 Article ID 012106 2011

[31] J Canny ldquoA computational approach to edge detectionrdquo IEEETransactions on Pattern Analysis and Machine Intelligence vol8 no 6 pp 679ndash698 1986

[32] A M Tartakovsky and P Meakin ldquoA smoothed particlehydrodynamics model for miscible flow in three-dimensionalfractures and the two-dimensional Rayleigh-Taylor instabilityrdquoJournal of Computational Physics vol 207 no 2 pp 610ndash6242005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics


ProbabilityandStatistics

Journal of



Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of



OptimizationJournal of


Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in



Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society


Advances in

DecisionSciences


Discrete MathematicsJournal of


Volume 2014


Stochastic AnalysisInternational Journal of


and surface tension forces For the chemical potential Oku-zono used the free energy of the system of Ginzburg-Landautype approximation with an order parameter Unlike thepresent approach Okuzono avoids calculating the secondderivative of the density explicitly instead he used the firstderivative two times in a row The results of Okuzono are ingood agreement with the power law for coarsening dynamicsThemodelrsquos major disadvantage is that it only describes phaseseparation if a phase distribution is known a priori

Thieulot et al [21] developed another phase separationmodel for a binary mixture It is based on the GENERICframework They derived the chemical potential directlyfrom the entropy of the fluid It takes the same shapeoriginally suggested by Cahn and Hilliard Their model isthermodynamically consistent but the effects of diffusivematerial transport to viscous and inertia forces are neglected

In this paper we develop a phase separation modelthat describes the diffusion-controlled stage using the Cahn-Hilliard equation It can easily be extended to describe allthree stages of coarsening by combination with existing SPHmodels As a prerequisite we introduce an approach tocalculate the fourth-order derivative in SPH

Most models use periodic boundary conditions Butthe effects of boundary conditions on the solution of theCahn-Hilliard equation are not negligible Therefore wediscuss different kinds of boundary conditions Since anextension tomulticomponentmixtures is straightforward weconsider only a binary fluid mixture in diffusion-controlledstage Analog to [9] one can extend the model for viscous-controlled and inertia-controlled stages

In Section 2 we first review the Cahn-Hilliard equationSubsequently we introduce the SPH model and an approxi-mation of the fourth-order derivative After that we discussdifferent kinds of boundary conditions In Section 3 we vali-date the approach considering a diffusion-controlled binarymixture with periodic boundary conditions Afterwards wedemonstrate the influence of different kinds of boundaryconditions on the solution of the phase separationmodel andconclude in Section 4

2 Materials and Methods

Themodel describes phase separation on amesoscaleThere-fore wemake one fundamental assumptionWe assume com-pleted nucleation on microscopic scale for the descriptionwith a continuummethod The initial state of the system willcontain uniformly distributed nuclei in space

First we review a phase separation model based on theCahn-Hilliard equation Next we introduce the SmoothedParticle Hydrodynamics method (SPH) and an approxima-tion for the fourth-order derivative Then we apply the SPHmethod to the phase separation model For simplicity weneglect viscous and inertia effects (ie the momentum bal-ance)That is valid in case of equalmolar volume density andnegligible excess values Finally since boundary conditionshave an important influence on the solution we focus on freesurfaces and solid walls

21 Phase Separation Model Phase decomposition is theresult of phase changes as it occurs in for example immer-sion precipitation condensation and evaporation A changeof temperature or concentration of a mixture can cause phasedecomposition After exceeding the binodal of the mixturea fluid mixture switches from stable (miscible) to metastableor unstable (immiscible) state respectively In an immisciblesystem nucleation occurs spontaneously after exceeding thespinodal The nuclei grow or disappear depending on theirchemical potentials After infinite time the system reachesits equilibrium state with homogeneous chemical potential inspace

A phase separation model describes the dynamics ofan immiscible fluid mixture based on fundamental ther-modynamics The thermodynamics describes the total freeenergy of the system It is the same energy on microscopicmesoscopic and macroscopic scale

In the late 1950s Cahn andHilliard [1] developed amodelbased on statistical thermodynamics and phenomenologicalobservations The model describes binary phase decompo-sition very well They start with a Taylor series expansionof the free energy density around the homogeneous stateTheir basic assumption was that the free energy density notonly depends on the local concentration but also on theconcentration of the immediate environment That meansthat they do not neglect higher order gradients of theconcentration As shown later by Fife [8] terms of higherorder are necessary for a stable solution in Hilbert space

The free energy results from Legendre transformation ofthe internal energy 119880

119865 = 119880 minus 119879119878 (1)

with 119879 and 119878 as temperature and entropy The free energy isalso

119865 = int119881

119891 (119888 nabla119888 nabla2119888 ) 119889119881 (2)

119881 is the volume of the system We assume an isothermal andisobaric systemThus the free energy density119891 only dependson the concentration 119888 and its derivatives As a result of thework of Cahn and Hilliard [1] we use

119891 (119888 nabla2119888) = 119891

0 (119888) +1

2120581(nabla119888)

2 (3)

to describe the free energy density With (3) (2) yields

119865 = int119881

(1198910 (119888) +

1

2120581(nabla119888)

2)119889119881 (4)

The first term is the free energy density of the homogeneousmixture It depends only on the local concentration Cahnand Hilliard supposed using

1198910 (119888) = prod

119894

(120596 (119879 119888119894) 119888119894) + 119896119861119879sum

119894

119888119894ln (119888119894) (5)

119896119861and119879 are Boltzmann constant and temperature 119894 indicates

the component







ℎ The weighting


119860 (x) = int119881ℎ

119860(x1015840)119882(ℎ x minus x1015840) 119889x1015840 (6)

int119881ℎ


0and 119871

0is the



119860 (x119894) asymp sum

119895

119898119895

120588119895

119860(x119895)119882(ℎ 119909

119894119895) (7)




119882(ℎ 119909119894119895)


120588119894= sum

119895

119898119895119882119894119895 (8)



120588119894= sum

119895

119898119895

119882119894119895

119882119894119895119904

119882119894119895119904

= sum

119895

119881119895119882119894119895

(9)


derivative


119895

119898119895

120588119895

(120572119894+ 120572119895)

x119894119895

1199092

119894119895






2(nabla2119860) (11)



119895

119898119895

120588119895

(120573119894+ 120573119895)

x119894119895

1199092

119894119895


with

11986010158401015840


119895

119898119895

120588119895

(120572119894+ 120572119895)

x119894119895

1199092

119894119895


(13)



Manzari [29]



leads to

119864119894= 4nabla119860|119894sum

119895

120596119895

1

119909119894119895

(sum

119895



119895

120596119895

1

119909119894119895

(sum

119895

120596119895e119894119895x119894119895e119894119895sdot nabla119894119882119894119895)



119895119896


(15)

e119894119895and 120596


119894119895119909119894119895and 119898

119895120588119895 The leading




|nabla119860|119894 times (119874(ℎ4(Δ

ℎ)

2120577+2

)) (16)


|nabla119860|119894 times (119874(1198892

119894Δ2(Δ

ℎ)

2120577+2

)) (17)




119863120588

119863119905= 0 (18)

119863119888

119863119905= nabla sdot (119872nabla120583) (19)



119894

120583119894=120575119865

120575119888119894

(20)


119863119888119894

119863119905= sum

119895

119898119895

120588119895

(119872119894+119872119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (120583119894minus 120583119895) (21)

with

120583119894=120597119891

120597119888119894


120597nabla119888119894

= 120596 (1 minus 2119888119894) + 119896119861119879 (ln (119888

119894) minus ln (1 minus 119888

119894)) minus

1

2120581nabla2119888119894

120581nabla2119888119894= sum

119895

119898119895

120588119895

(120581119894+ 120581119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (119888119894minus 119888119895)

(22)







119865 = int119881

119891 (119888 nabla2119888) 119889119881 + int

Ω

119891119908 (119888) 119889Ω (23)



intΩ

119891119908 (119888) 119889Ω =

int119881119891119908 (119888) 119889119881

1198710

(24)


119908(119888) It can be

written as

120575120583119908 (119888) = minus

120574 (12059012 1205901119908 1205902119908 119888)

21198882 cos120572 (25)


119861119879 = 021 120581 = 2)





Ω (119905) prop 119905minus13

(26)





0=








120598 (119905) =(ΩSPH (119905) minus ΩFD (119905))

2

ΩFD(119905)2

(27)






n sdot nabla1205831003816100381610038161003816120597Ω = 0 (28)





119880 gt 119910 (29)



1198880=

1

4120587= 00796

1 minus1

4120587= 09204

(30)


0lt 09204







(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

104

103

102

104

103

102

Time log(t)

Surfa

ce ar

ealog(Ω)


FDPower law

(a)

100

10minus1

10minus2

10minus3

10minus4

Erro

rlog(t)

104

103

102

Time log(t)


(b)



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)



119905 = 50 (h) and 119905 = infin (i)



0and Δ119888



0= 099 and of the

larger one 1198880= 001Δ119888



∘ and120572 = 180


∘ and 120572 = 180∘



(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014









ℎ The weighting


119860 (x) = int119881ℎ

119860(x1015840)119882(ℎ x minus x1015840) 119889x1015840 (6)

int119881ℎ


0and 119871

0is the



119860 (x119894) asymp sum

119895

119898119895

120588119895

119860(x119895)119882(ℎ 119909

119894119895) (7)




119882(ℎ 119909119894119895)


120588119894= sum

119895

119898119895119882119894119895 (8)



120588119894= sum

119895

119898119895

119882119894119895

119882119894119895119904

119882119894119895119904

= sum

119895

119881119895119882119894119895

(9)


derivative


119895

119898119895

120588119895

(120572119894+ 120572119895)

x119894119895

1199092

119894119895






2(nabla2119860) (11)



119895

119898119895

120588119895

(120573119894+ 120573119895)

x119894119895

1199092

119894119895


with

11986010158401015840


119895

119898119895

120588119895

(120572119894+ 120572119895)

x119894119895

1199092

119894119895


(13)



Manzari [29]



leads to

119864119894= 4nabla119860|119894sum

119895

120596119895

1

119909119894119895

(sum

119895



119895

120596119895

1

119909119894119895

(sum

119895

120596119895e119894119895x119894119895e119894119895sdot nabla119894119882119894119895)



119895119896


(15)

e119894119895and 120596


119894119895119909119894119895and 119898

119895120588119895 The leading




|nabla119860|119894 times (119874(ℎ4(Δ

ℎ)

2120577+2

)) (16)


|nabla119860|119894 times (119874(1198892

119894Δ2(Δ

ℎ)

2120577+2

)) (17)




119863120588

119863119905= 0 (18)

119863119888

119863119905= nabla sdot (119872nabla120583) (19)



119894

120583119894=120575119865

120575119888119894

(20)


119863119888119894

119863119905= sum

119895

119898119895

120588119895

(119872119894+119872119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (120583119894minus 120583119895) (21)

with

120583119894=120597119891

120597119888119894


120597nabla119888119894

= 120596 (1 minus 2119888119894) + 119896119861119879 (ln (119888

119894) minus ln (1 minus 119888

119894)) minus

1

2120581nabla2119888119894

120581nabla2119888119894= sum

119895

119898119895

120588119895

(120581119894+ 120581119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (119888119894minus 119888119895)

(22)







119865 = int119881

119891 (119888 nabla2119888) 119889119881 + int

Ω

119891119908 (119888) 119889Ω (23)



intΩ

119891119908 (119888) 119889Ω =

int119881119891119908 (119888) 119889119881

1198710

(24)


119908(119888) It can be

written as

120575120583119908 (119888) = minus

120574 (12059012 1205901119908 1205902119908 119888)

21198882 cos120572 (25)


119861119879 = 021 120581 = 2)





Ω (119905) prop 119905minus13

(26)





0=








120598 (119905) =(ΩSPH (119905) minus ΩFD (119905))

2

ΩFD(119905)2

(27)






n sdot nabla1205831003816100381610038161003816120597Ω = 0 (28)





119880 gt 119910 (29)



1198880=

1

4120587= 00796

1 minus1

4120587= 09204

(30)


0lt 09204







(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

104

103

102

104

103

102

Time log(t)

Surfa

ce ar

ealog(Ω)


FDPower law

(a)

100

10minus1

10minus2

10minus3

10minus4

Erro

rlog(t)

104

103

102

Time log(t)


(b)



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)



119905 = 50 (h) and 119905 = infin (i)



0and Δ119888



0= 099 and of the

larger one 1198880= 001Δ119888



∘ and120572 = 180


∘ and 120572 = 180∘



(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014





|nabla119860|119894 times (119874(ℎ4(Δ

ℎ)

2120577+2

)) (16)


|nabla119860|119894 times (119874(1198892

119894Δ2(Δ

ℎ)

2120577+2

)) (17)




119863120588

119863119905= 0 (18)

119863119888

119863119905= nabla sdot (119872nabla120583) (19)



119894

120583119894=120575119865

120575119888119894

(20)


119863119888119894

119863119905= sum

119895

119898119895

120588119895

(119872119894+119872119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (120583119894minus 120583119895) (21)

with

120583119894=120597119891

120597119888119894


120597nabla119888119894

= 120596 (1 minus 2119888119894) + 119896119861119879 (ln (119888

119894) minus ln (1 minus 119888

119894)) minus

1

2120581nabla2119888119894

120581nabla2119888119894= sum

119895

119898119895

120588119895

(120581119894+ 120581119895)

x119894119895

1199092

119894119895

nabla119894119882119894119895sdot (119888119894minus 119888119895)

(22)







119865 = int119881

119891 (119888 nabla2119888) 119889119881 + int

Ω

119891119908 (119888) 119889Ω (23)



intΩ

119891119908 (119888) 119889Ω =

int119881119891119908 (119888) 119889119881

1198710

(24)


119908(119888) It can be

written as

120575120583119908 (119888) = minus

120574 (12059012 1205901119908 1205902119908 119888)

21198882 cos120572 (25)


119861119879 = 021 120581 = 2)





Ω (119905) prop 119905minus13

(26)





0=








120598 (119905) =(ΩSPH (119905) minus ΩFD (119905))

2

ΩFD(119905)2

(27)






n sdot nabla1205831003816100381610038161003816120597Ω = 0 (28)





119880 gt 119910 (29)



1198880=

1

4120587= 00796

1 minus1

4120587= 09204

(30)


0lt 09204







(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

104

103

102

104

103

102

Time log(t)

Surfa

ce ar

ealog(Ω)


FDPower law

(a)

100

10minus1

10minus2

10minus3

10minus4

Erro

rlog(t)

104

103

102

Time log(t)


(b)



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)



119905 = 50 (h) and 119905 = infin (i)



0and Δ119888



0= 099 and of the

larger one 1198880= 001Δ119888



∘ and120572 = 180


∘ and 120572 = 180∘



(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014





Ω (119905) prop 119905minus13

(26)





0=








120598 (119905) =(ΩSPH (119905) minus ΩFD (119905))

2

ΩFD(119905)2

(27)






n sdot nabla1205831003816100381610038161003816120597Ω = 0 (28)





119880 gt 119910 (29)



1198880=

1

4120587= 00796

1 minus1

4120587= 09204

(30)


0lt 09204







(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

104

103

102

104

103

102

Time log(t)

Surfa

ce ar

ealog(Ω)


FDPower law

(a)

100

10minus1

10minus2

10minus3

10minus4

Erro

rlog(t)

104

103

102

Time log(t)


(b)



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)



119905 = 50 (h) and 119905 = infin (i)



0and Δ119888



0= 099 and of the

larger one 1198880= 001Δ119888



∘ and120572 = 180


∘ and 120572 = 180∘



(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2

104

103

102

104

103

102

Time log(t)

Surfa

ce ar

ealog(Ω)


FDPower law

(a)

100

10minus1

10minus2

10minus3

10minus4

Erro

rlog(t)

104

103

102

Time log(t)


(b)



(a) (b) (c)

(d) (e) (f)

(g) (h) (i)



119905 = 50 (h) and 119905 = infin (i)



0and Δ119888



0= 099 and of the

larger one 1198880= 001Δ119888



∘ and120572 = 180


∘ and 120572 = 180∘



(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




(a) (b) (c)

(d) (e) (f)

(g) (h) (i)



119905 = 50 (h) and 119905 = infin (i)



0and Δ119888



0= 099 and of the

larger one 1198880= 001Δ119888



∘ and120572 = 180


∘ and 120572 = 180∘



(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




(a) (b) (c)

(d) (e) (f)


∘ (c) 120572 = 90∘

(d) 120572 = 120∘ (e) and 120572 = 180








4 Conclusions








(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




(a) (b) (c)

(d) (e) (f)


0= 04

120596 = 1 119896119861119879 = 021 and 120581 = 2





Acknowledgment


References









































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014




































Volume 2014




Journal of



Volume 2014

Advances in







Volume 2014


Combinatorics

OperationsResearch

Advances in









Algebra



Advances in

DecisionSciences




Volume 2014



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