cahn–hilliard modeling of particles suspended in two-phase ﬂows · 2011-06-24 · ticles...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids (2011)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2623

Cahn–Hilliard modeling of particles suspended in two-phase flows

Young Joon Choi and Patrick D. Anderson*,†

Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

SUMMARY

In this paper, we present a model for the dynamics of particles suspended in two-phase flows by couplingthe Cahn–Hilliard theory with the extended finite element method (XFEM). In the Cahn–Hilliard model theinterface is considered to have a small but finite thickness, which circumvents explicit tracking of the inter-face. For the direct numerical simulation of particle-suspended flows, we incorporate an XFEM, in which theparticle domain is decoupled from the fluid domain. To cope with the movement of the particles, a temporaryALE scheme is used for the mapping of field variables at the previous time levels onto the computationalmesh at the current time level. By combining the Cahn–Hilliard model with the XFEM, the particle motionat an interface can be simulated on a fixed Eulerian mesh without any need of re-meshing. The model isgeneral, but to demonstrate and validate the technique, here the dynamics of a single particle at a fluid–fluid interface is studied. First, we apply a small disturbance on a particle resting at an interface between twofluids, and investigate the particle movement towards its equilibrium position. In particular, we are interestedin the effect of interfacial thickness, surface tension, particle size and viscosity ratio of two fluids on theparticle movement towards its equilibrium position. Finally, we show the movement of a particle passingthrough multiple layers of fluids. Copyright © 2011 John Wiley & Sons, Ltd.

Received 25 February 2011; Revised 3 May 2011; Accepted 8 May 2011

KEY WORDS: Cahn–Hilliard theory; diffuse-interface model; extended finite element method (XFEM);temporary ALE scheme; finite element; hydrodynamics; incompressible flow; laminar flow;two-phase flows

1. INTRODUCTION

Small solid particles adsorbed at liquid interfaces arise in many industrial products and processes,such as antifoam formulations, crude oil emulsions, fluidized suspensions, slurry transport, materialsseparation, rate of mixing enhancement, etc. In particular, if particles are suspended in emulsions,this emulsion can be stabilized by solid particles that adsorb onto the interface between the twofluids, which is usually called a Pickering emulsion [1]. They act in many ways like traditionalsurfactant molecules, but offer distinct advantages. Unfortunately, the understanding of how theseparticles operate in such systems is limited. In this paper, we present a numerical method for thedynamics of particles in two-phase flows based on the Cahn–Hilliard theory.

Diffuse-interface modeling is based on the van der Waals approach of the interface problem [2]and developed by Cahn and Hilliard [3]. The main assumption is that the interface is not sharp,but has a thickness that is not explicitly prescribed, but follows from the governing equations thatcouple thermodynamic and hydrodynamic forces. The main elements of the theory, and the couplingof thermodynamics and hydrodynamics are summarized in the review paper by Anderson et al. [4]and references therein.

*Correspondence to: Patrick D. Anderson, Department of Mechanical Engineering, Eindhoven University of Technology,PO Box 513, 5600 MB Eindhoven, The Netherlands.

†E-mail: [email protected]

Copyright © 2011 John Wiley & Sons, Ltd.

Y. J. CHOI AND P. D. ANDERSON

Diffuse-interface methods have been applied to a variety of multiphase flow problems rangingfrom phase separating polymer blends to simulating solid tumor growth using mixture models.For example, Prusty et al. used the Cahn–Hilliard technique to the coarsening dynamics forPMMA/SAN28 blends and a quantitative comparison between the experimental and numericallypredicted phase separation kinetics was presented [5]. Wise et al. presented simulations of tumorgrowth in two and three dimensions that demonstrate the capabilities of the diffuse-interface modelin accurately and efficiently simulating the progression of tumors with complex morphologies [6].Khatavkar et al. used the diffuse-interface method to model micron-sized drop spreading and impacton smooth and structured substrates [7–9]. Recently, Tufano et al. applied the Cahn–Hilliard theorycoupled with hydrodynamic interactions to describe a three-phase systems where the effects ofmutual diffusion on interfacial tension, drop–drop and drop–wall interactions in quiescent condi-tions are investigated and compared with experimental observations [10]. Millet and Wang intro-duced a diffuse-interface field description of each fluid phase in addition to the set of solid particles.Their model can include particles of arbitrary shapes and orientations, and the ability to incorporateelectrostatic particle interactions [11].

The most intuitive method to simulate particle movement in two-phase flows is using a boundary-fitted mesh, which means that the particle surface is aligned with element boundaries of thefluid [12–14]. In this method, the governing equations are solved only in the fluid domain, takinginto account the interface conditions on the boundaries of particles. To handle moving particles, thisapproach incorporates the ALE technique that relies on a moving mesh scheme. The generation ofa new mesh is needed if the old mesh becomes too distorted, and the solution must be projectedonto the new mesh. The generation of boundary-fitted meshes is, however, still a challenging task inview of algorithms needed and computational costs if complex geometries are involved, especiallyin three-dimensional simulations using second-order hexahedron elements.

An alternative approach is the fictitious domain method developed by Glowinski et al. [15–17].The basic idea of this method is to fill each domain of every particle with the surrounding fluid,assuming and subsequently prescribing that the fluid inside the particle domain moves like a solidobject. Hence, the problem is transferred from a geometrically complex fluid domain to a simplerdomain including both fluid and particles, which eliminates the need of re-meshing. In this method,particles move in a Lagrangian sense on a fixed Eulerian mesh. For single-component problems thisapproach has been quite successful and the dynamics of particles in complex fluids has been studiedin a variety of flow conditions [18, 19]. However, if a particle is suspended in a two-componentsystem, a fictitious domain approach would require additional constraints for the motion of fic-titious fluids inside the particle, which is not trivial and in this work an alternative approach isfollowed.

The extended finite element method (XFEM) has been recently developed to simulate particlesuspended single-component fluid flows. In this method, the finite element shape functions arelocally extended, or enriched, to decouple the fluid domain from the particle domain while stillusing a mesh that is not boundary fitted. Originally, XFEM was developed for the simulations ofcracks in structures without the need of re-meshing [20,21]; later, it was applied to flow problems aswell [22–24]. A recent review on XFEMs applied to material modeling is presented in [25]. Choiet al. proposed a temporary arbitrary Lagrangian–Eulerian (tALE) scheme to handle moving par-ticles without any need of re-meshing throughout the whole computations in the extended finiteelement context [26].

In the present paper, we present a numerical method for the dynamics of particles in two-phase flows by coupling the Cahn–Hilliard theory with the XFEM while using a fixed Eulerianmesh without any need of re-meshing. Because the fluid domain is decoupled from the parti-cle domain in XFEM, we do not need extra conditions inside the particles, contrary to the fic-titious domain method. The content of this paper is as follows. In Section 2 we give a briefreview of the Cahn–Hilliard theory. In Section 3 the numerical algorithm of the XFEM combinedwith a tALE scheme is discussed. The introduced model is applied to study the dynamics of asingle particle at a fluid–fluid interface in Section 4, and the movement of a particle passingthrough multiple layers of fluids is demonstrated in Section 5. Finally, a discussion follows inSection 6.

Copyright © 2011 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2011)DOI: 10.1002/fld

CAHN–HILLIARD MODELING OF PARTICLES SUSPENDED IN TWO-PHASE FLOWS

2. MATHEMATICAL FORMULATION

The classical expression for the specific Helmholtz free energy used in diffuse interface modelingis based on the work of Cahn and Hilliard [3]

f .c,rc/D�1

2˛c2C

1

4ˇc4C

1

2" jrcj2 , (1)

where ˛ and ˇ are positive constants and " is the gradient-energy parameter and c is the mass frac-tion of one of the two components [27]. The chemical potential is obtained from the variationalderivative with respect to concentration

�Dıf

ıcD�˛cC ˇc3 � "r2c. (2)

This generalized chemical potential allows for the description of the interface between the twomaterials by a continuously varying concentration profile. For example, for a planar interface, withx being the direction normal to the interface, the analytical solution of Equation (2) reads

c.x/D

r˛

ˇtanh

xp2�

, (3)

with ˙p˛=ˇ the equilibrium bulk solutions (in the approach outlined here ˙1) and � (D

p"=˛),

which is a measure for the interfacial thickness. In order to comply with mass conservation for bothcomponents, the balance equation should be fulfilled

@c

@tC u � rc Dr � .Mr�/, (4)

withM the mobility, which in general is a function of the composition, but is here taken constant forsimplicity. The diffusion flux is assumed to be proportional to the gradient of the chemical poten-tial, which is more general than the common Fickian diffusion, based on the concentration gradients(rc), which does not hold for multiphase systems, even at equilibrium. The more general expressionused in Equation (4) reflects Gibbs’ findings that the chemical potential becomes uniform in a non-ideal mixture at equilibrium, and is known as the Cahn–Hilliard equation [28]. To obtain momentumconservation, a generalized Navier�Stokes equation can be derived for the velocity field [29]

�

�@u

@tC .u � r/u

�D��rgCr � .2�D/C ��rc, (5)

r � uD 0, (6)

whereD D .ruCruT /=2 is the rate-of-deformation tensor; g is the Gibbs free energy gDfCp=�,with p the local pressure and � the density. The viscosity � generally depends on c because the twofluids, in general, have different viscosities. The viscosity �, is assumed to have the following linearrelationship with the concentration c,

�D �1

�cC 1

2

�� 2

�c � 1

2

�, (7)

where �1 and �2 are the viscosities of the two fluids, respectively.Compared with the Navier–Stokes equations, in Equation (5) only one extra capillary term ��rc

appears reflecting the interfacial tension. This modification accounts for hydrodynamic interactions,that is, the influence of the concentration c or the morphology on the velocity field and, hence,describes the spatial variations of the velocity field because of the presence of interfaces.

To focus on the coupling of the particles with the multiphase system, without any loss of gen-erality, we now further assume quasi-stationary flow and neglect inertia in the momentum balance.Then, the momentum balance Equation (5) reduces to

�r � .2�D/Crg D ��rc. (8)



2.1. Scaling of the equations

Using c� D c=cB , u� D u=U , �� D ��2=."cB/, t� D tU=L, with cB Dp˛=ˇ the bulk con-

centration, U a characteristic velocity, and L a characteristic domain size, and omitting the asterisknotation, Equations (2), (4), (6), and (8) read in dimensionless form

dc

dtD

1

Per2�, (9)

�D c3 � c �C 2r2c, (10)

�r � .2�D/Crg D1

CaC�rc, (11)

r � uD 0. (12)

Three dimensionless groups appear in the governing equations: Péclet number Pe, the capillarynumber Ca and the Cahn number C , defined as

Pe D�2LU

M"I CaD

��U

�"c2BI C D

�

L. (13)

2.2. Rigid-body motion of particles

We suppose that N particles are suspended in an incompressible fluid. Let � be the entire domainincluding the fluid and particles, and let Pi .t/ .i D 1, : : : ,N/ be the embedded domain of thei-th particle at time t with the number of particles N . The collective particle region at a certaintime t is denoted by P.t/ D

SNiD1 Pi .t/. Boundaries are denoted by � D @� and @Pi .t/. For the

unknown rigid body motions (U i , !i ) of the particles, we need balance equations for forces andtorques on particle surfaces. In the absence of inertia, external forces Fext,i and torques Text,i actingon the particle Pi .t/ are balanced by the net hydrodynamic force Fi and torque Ti on the particle,respectively

Fi D

Z@Pi .t/

� � n ds D Fext,i , (14)

Ti D

Z@Pi .t/

.x �X i /� .� � n/ ds D Text,i , (15)

where n is the outwardly-directed unit normal vector on the particle surface @Pi .t/. The particlepositions X i and angular orientations‚i are obtained from the following kinematic equations:

dX i

dtD U i , X i .t D 0/DX i ,0, (16)

d‚i

dtD!i , ‚i .t D 0/D‚i ,0. (17)

At the fluid–particle interface, we use the no-slip boundary condition

uD U i C!i � .x �X i / on @Pi .t/.i D 1, : : : ,N/. (18)

2.3. Partial wetting boundary conditions

The particle may be neutral or preferably wetted by one of the components of the binary fluid. Thiseffect is accounted for by following the approach given by Cahn [30], where solid–fluid interactions



are assumed to be short ranged. Because of this assumption, the total system free energy F can bewritten as

F DZ�nP.t/

f dV CNXiD1

Z@Pi .t/

fp ds, (19)

where fp is the specific particle free energy that depends only on the concentration at the particleand �nP.t/ is the fluid domain volume bounded with a particle surface @P.t/ D

SNiD1 @Pi .t/ and

f is defined in Equation (1). The surface integral term in Equation (19) represents the contributionof solid–fluid interactions.

At equilibrium, F is at its minimum. Minimizing F using methods of variational calculus subjectto a natural boundary condition gives the following boundary condition on the particle surface@Pi .t/:

� �@c

@nC@fp

@cD 0, (20)

where n is the direction normal to @Pi .t/ and fp is the specific particle free energy.For fp we use the form proposed by Jacqmin [31], which reads,

fp D

�c �

c3

3

�, (21)

where is assumed to be a constant and referred to as the wetting potential. It can be made tovary spatially to indicate chemical heterogeneity of the particle surface. With fp of the form (21),@fp=@c evaluated at cB is zero, so at equilibrium the particle surface is not enriched in one of thefluids and depleted in the other.

Equations (20) and (21) are nondimensionalized using the dimensionless variables defined inSection 2.1 with the addition of lv as the characteristic scale for the specific particle free energy togive

�C@c

@nC

�1� c2

�D 0, (22)

fp D

�c �

c3

3

�, (23)

where D .�cB/=.lv/ is the dimensionless wetting potential and C is the Cahn number definedin Equation (13).

Using Young’s equation, which connects the contact angle � with the interfacial tensions lv , svand sl

cos � Dsv � sl

lv, (24)

which is also shown in Figure 1. The parameter can be related to the (equilibrium) contact angle� to yield

cos � D4

3. (25)

From (25) it is concluded that for a contact angle � equal to 90ı, is zero and the mixed boundarycondition (22) reduces to the natural boundary condition @c=@nD 0.



Fluid 2

Fluid 1

Particle

Figure 1. Definition of contact angle � .

3. NUMERICAL METHODS

3.1. Weak form

In deriving the weak form of the governing equations for a fluid–particle system, we follow the com-bined equation of motion approach [17], in which the hydrodynamic forces and torques acting onparticles are eliminated from the equation of motion because they are internal. The no-slip boundarycondition on the particle surface is imposed by using constraints implemented with Lagrangianmultipliers. �

r ,@c

@tC u � rc

�CM .rr ,r�/D 0, (26)

�e,˛c � ˇc3

�� " .re,rc/C .e,�/D 0, (27)

�.rv/T , 2�D

�� .r � v,g/C .v� .V i C�i � .x �X i //,�i /@Pi .t/D .v, t/�N

C V i �F ext,i C�i � T ext,i C .v, ��rc/ , (28)

.q,r � u/D 0, (29)

.�i ,u� .U i C!i � .x �X i ///@Pi .t/ D 0, (30)

for all admissible test functions r , e, v, q,�i , V i and�i . .�, �/, .�, �/�N , and .�, �/@Pi .t/ are proper innerproducts on the fluid domain �nP.t/, on the Neumann boundary �N and on the particle interface@Pi .t/, respectively.

3.2. Time discretization of the diffuse-interface model

We solve the governing Equations (26)–(30) in a decoupled way. First, the concentration c andchemical potential � are solved simultaneously from Equations (26) and (27) with appropriateboundary conditions. Then, we solve the Stokes-type flow problem by treating the additional ��rcterm as a forcing.

For the time discretization of the evolution equation of the concentration (Equation (26)), we usea second-order Gear scheme

r ,32cnC1 � 2cnC 1

2cn�1

tC OunC1 � rcnC1

!CM

�rr ,r�nC1

�D 0, (31)

�e,˛cnC1 � ˇ.cnC1/3

�� "

�re,rcnC1

�C�e,�nC1

�D 0, (32)

where t is the time step and OunC1 D 2.un � unm/ � .un�1 � un�1m /. Here um represents a mesh

velocity because of a mesh movement scheme that will be defined in Section 3.4. Because theseequations are nonlinear, we solve them by Picard iteration at each time level t D tnC1



�r ,3

2

ciC1

tC OunC1 � rciC1

�CM .rr ,r�iC1/D

r ,2cn � 1

2cn�1

t

!, (33)

�e, .˛ � ˇc2i /ciC1

�� " .re,rciC1/C .e,�iC1/D 0, (34)

for i D 0, 1, : : : until convergence, with c0 D cn. For the first time step t D t , we use an implicitEuler scheme instead of Equation (31)�

r ,cnC1 � cn

tC .un � unm/ � rc

nC1

�CM

�rr ,r�nC1

�D 0. (35)

The complete time integration steps are given in Section 3.5.

3.3. XFEM formulation

For the direct numerical simulation of flows with freely suspended particles, we use the XFEMcombined with the tALE scheme, which was presented by Choi et al. [26]. Here we briefly reviewthe method. For a detailed description of the method, please see [26] and the references therein.

In the XFEM context, the velocity, pressure, concentration, and chemical potential can bediscretized as

uh.x/DXi

�i .x/H.s/ui , (36)

gh.x/DXi

i .x/H.s/gi , (37)

ch.x/DXi

�i .x/H.s/ci , (38)

�h.x/DXi

�i .x/H.s/�i , (39)

where

H.s/D

²C1 if s > 0,0 if s < 0,

(40)

is defined by a level set function s

s.x/

8<:D 0 if x is on @P.t/,> 0 if x is in �nP.t/,< 0 if x is in P.t/.

(41)

We use a biquadratic Q2 interpolation (�i ) for the velocity u, concentration c, and chemicalpotential �; a bilinear Q1 interpolation ( i ) for the modified pressure g.

For elements intersected by the surface of a particle, the no-slip boundary condition (Equation(18)) is imposed by using constraints implemented with Lagrangian multipliers as shown inEquations (28) and (30). The inner product .�, �/@Pi .t/ is the standard inner product in L2.@Pi .t//:

.�i ,u� .U i C!i � .x �X i ///@Pi .t/ D

Z@Pi .t/

�i � .u� .U i C!i � .x �X i /// ds. (42)

For the discretization of Equation (42), we use a linear shape function P1 for the discretization ofLagrangian multipliers and a quadratic shape function P2 for the geometrical shape of each element.



3.4. Temporary ALE scheme

For a moving particle problem, the field variable at the previous time levels, such as un, un�1, cn

and cn�1, can become undefined near the boundary of the particle because there may have beenno fluid flow at time level tn. To overcome this problem, we use a tALE scheme, which defines amapping of field variables at the previous time levels on the current time level [26].

In this method, mesh nodes near a particle follow the motion of the particle, whereas, meshnodes far away from the particle are stationary. A mesh velocity field um is solved using Laplace’sequation:

r2um D 0 in �, (43)

um D 0 on � , (44)

um D U i C!i � .x �X i / on @Pi .t/. (45)

For a circular particle, Equation (45) can be replaced by um D U i .Note that Equations (43)–(45) are solved on an Eulerian mesh �, including the particle domain

P.t/, by using a similar technique as in the fictitious domain method [15–17]. Equation (45) isrealized by using a constraint implemented with Lagrangian multipliers.

An ALE mesh at the previous time level t D tn, xnALE, is constructed using a predictor–correctormethod

x�m D xn�1m C um.x

n�1m , tn�1/ t (predictor) (46)

xnALE D xn�1m C

1

2

�um.x

�m, tn/C um.x

n�1m , tn�1/

t (corrector) (47)

Then, the ALE mesh at the current time level, xnC1ALE , can be constructed using a second-orderAdams–Bashforth method (AB2):

xnC1ALE D xnALEC

�3

2um.x

nALE, tn/�

1

2um.x

n�1m , tn�1/

� t (AB2) (48)

Equations (46) and (47) define the mapping ‰ and Equation (48) defines the mapping ˆ. Themappings‰ andˆ are represented in Figure 2. The field variables at previous time levels are mappedalong with the ALE meshes (Figure 3):

cn D c.ˆ�1.x/, tn/, (49)

un D u.ˆ�1.x/, tn/, (50)

unm D um.ˆ�1.x/, tn/, (51)

cn�1 D c.‰�1 ıˆ�1.x/, tn�1/, (52)

un�1 D u.‰�1 ıˆ�1.x/, tn�1/, (53)

un�1m D um.‰�1 ıˆ�1.x/, tn�1/. (54)

Note that the unknowns at the current time level, such as cnC1 and �nC1, are computed on the fixedEulerian mesh.

3.5. Time integration

At the initial time t D 0we solve the flow equations without the right-hand side term ��rc to obtainan initial flow solution. Also, the initial concentration field c0 D c.t D 0/ should be specified. Then,we apply the following procedure at every time step:

Step 1. Construct a temporary ALE mesh using Equations (46)–(48) for the interpolation of fieldvariables at previous time levels. At the first time step, we use a first-order scheme givenby

xnC1ALE D xnmC um.x

nm, tn/ t . (55)



at t = tn −1 at t = tn at t = tn +1

(a) n −1m (b) n

m

(c) nALE

(d) n +1m = n

m = n − 1m

(e) n +1ALE

x

x x

x x x x

Figure 2. Construction of tALE meshes xnALE and xnC1ALE using a second-order scheme, which defines themappings ‰ and ˆ. (a) xn�1m , (b) xnm, (c) xnALE, (d) xnC1m D xnm D x

n�1m , and (e) xnC1ALE .

(a) t = tn −1 (b) t = tn (c) t = tn +1

nn − 1

−1

−1x x

x

Figure 3. Field variables at previous time levels are mapped along with the ALE meshes. The advection ofthe ALE meshes defines the mappings ‰ and ˆ. (a) t D tn�1, (b) t D tn, and (c) t D tnC1.

Step 2. Update the particle configuration by integrating the kinematic equations in Equations (16)and (17) using the explicit second-order Adams–Bashforth method (AB2)

XnC1i DXn

i C

�3

2U ni �

1

2U n�1i

� t , (56)

‚nC1i D‚n

i C

�3

2!ni �

1

2!n�1i

� t . (57)

For circular particles, the update of angular rotations is not necessary. At the first timestep, we use an explicit Euler method

XnC1i DXn

i CUni t . (58)



Step 3. Modify the computational mesh to avoid very small integration areas [26]. First, computea mesh velocity field Oum

r2 Oum D 0 in �, (59)

Oum D 0 on � , (60)

Oum D n on @Pi .t/, (61)

where n is the outwardly-directed unit normal vector on the particle surface. Then, eachmesh point xm moves according to the following advection equations

dxm

dtmD

²Oum if xm 2�nP.t/,0 if xm 2 P.t/,

(62)

xm.tm D 0/D xm,0 (63)

where xm,0 is the initial position of mesh point. In our simulations, we use a third-order Adams–Bashforth method (AB3), and nodal points are moved until each area ofintegration is larger than 0.5% of the element area.

Step 4. Compute cnC1 and �nC1 by solving Equations (33) and (34) iteratively.Step 5. Compute unC1, gnC1, �nC1i , U nC1i , and !nC1i from the momentum balance and

continuity equation�.rv/T , 2�D.unC1/

��r � v,gnC1

�C�v� .V i C�i � .x �X

nC1i //,�nC1i

�@Pi .t

nC1/

D .v, t/�NC V i �F ext,i C�i � T ext,i C

�v, ��nC1rcnC1

�, (64)

�q,r � unC1

�D 0, (65)

��i ,u

nC1 � .U nC1i C!nC1i � .x �XnC1i //

�@Pi .t

nC1/D 0. (66)

Equations (33) and (34) are solved by using a direct solver HSL MA41, and Equations (64)–(66)are solved by using a direct symmetric solver HSL MA57 [32].

4. PARTICLE AT A FLUID–FLUID INTERFACE

4.1. Problem description

As a model problem, a particle is placed at a fluid–fluid interface, confined between two parallelplates. Initially, we assume the steady state condition, that is, the fluids are stationary and the parti-cle is at rest in the middle of the fluid–fluid interface (see Figure 4). The particle radius is denotedby a, the thickness of the fluid–fluid interface by � , and the viscosity of lower and upper fluids are�1 and �2, respectively. The effective viscosity � is assumed to have a linear relationship given byEquation (7). At t D 0, we disturb the flow by applying an external force F D .0,�Fy/ on theparticle in the y direction for a certain time duration tF . For t > tF , the external force F on theparticle is removed, and the particle freely moves to its equilibrium position as a result of the actingsurface tension forces. Note that dependent on the value of tF the contact position of the interfacewith the particle may change, but as long as the particle remains at the interface for t < tF it willreturn to its equilibrium position.

In this problem, the scaling of equations by dimensionless groups given in Equation (13) is nottrivial because a characteristic velocity U is unknown prior to solving the fluid and particle veloci-ties. Instead of stating the dimensionless groups given in Equation (13), we provide the actual valuesused in our simulations, and in principle one could determine the characteristic velocity U to esti-mate the magnitude of the characteristic dimensionless groups. We fix the channel heightH D 1 andchannel length LD 1, assuming that the top and bottom walls are stationary and the flow is periodicin the x direction. Otherwise stated, the particle radius equals a D 0.15, the viscosities �1D�2D1,



L

H

F

1

2

c = +1

c = −1

y

a

x

Figure 4. Geometry for a particle at a fluid–fluid interface at t D 0.

the magnitude of the external force Fy D 10, and ˛ D 1, ˇ D 1, "D 0.0001,M D 0.01, �D 100 formaterial parameters used in governing Equations (2), (4), and (8). Note that the interfacial thickness� D

p"=˛ D 0.01 for the given values.

Figure 5 shows the concentration profile at t D 0.1 for forcing time duration tF D 0.1 withFy D 10, where the particle position is lowest for the given forcing condition. After validation ofthe computational scheme, we will investigate the particle motion to its equilibrium position forvarious parameters - tF , a, �2=�1, and so on.

4.2. Convergence test

Before we study the effect of the different material and process parameters on the dynamics of theparticle, we first demonstrate the convergence of the method by mesh and time step refinements.Four meshes are defined with decreasing element size — M1 (70�70 elements), M2 (100�100elements), M3 (125�125 elements), and M4 (150�150 elements), and the mesh parameters aresummarized in Table I. Figure 6 shows the position of the center of the particle as a function of

Figure 5. Concentration contours at t D 0.1 with forcing time tF D 0.1 and Fy D 10.



Table I. Meshes used for the simulations.

Mesh Number of elements Number of nodes

M1 4900 19,881M2 10,000 40,401M3 15,625 63,001M4 22,500 90,601

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 10 20 30 40 50

t

M1M2M3M4

y

Figure 6. Mesh convergence showing the position of the particle as a function of time t 6 50 with forcingtime tF D 0.1.

time for the case of forcing time duration tF D 0.1 with the external force Fy D 10. The timestep t D 0.001 is used for all meshes. The result of M1 shows nonmonotonic behavior of particlemovement because of the unresolved mesh resolution; the results of M3 and M4 are fully overlappedand cannot be distinguished.

The time step convergence is checked by using mesh M3 as it demonstrated a fully resolved meshresolution. Figure 7 shows the histories of the particle position obtained by using various time steps.For t D 0.01, the particle motion is predicted slightly slower than the other cases. For all othercases, t 6 0.002, the particle positions fully overlap, but if we use a time step t > 0.02, thesimulation becomes unstable. Hence, for all our simulations in the paper, we use the mesh M3 incombination with the time step t D 0.001.

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 10 20 30 40 50

t

t = 0.01t = 0.002t = 0.001t = 0.0005

ΔΔΔΔ

y

Figure 7. Time convergence showing the position of the particle as a function of time t 6 50.



Finally, to obtain more information about the induced flow by the retracting particle, we study thevorticity of the flow as a function of time. Note that the vorticity is defined by

! D@v

@x�@u

@y, (67)

where u and v are the x-directional and y-directional velocity of the fluids, respectively. Figure 8shows the vorticity in the domain at times t D 1.0 and t D 5.0 with forcing time tF D 0.1. Thefigure shows that the magnitude of the vorticity is larger below the particle compared with above theparticle and a flow is induced to push the particle towards its equilibrium position. Figure 9 showsthe corresponding pressure plots at the same time levels; these figures also show a higher pressurebelow the particle that pushes the particle in the direction back to equilibrium. Figure 10 shows themaximum of the absolute value of the vorticity in the fluid domain j!jmax as a function of time.During 06 t 6 tF , the maximum vorticity increases because of external force F , then decreases tozero as time goes on. The maximum vorticity obtained by using M1 and M2 shows fluctuations intime especially when we pull down the particle during 0 6 t 6 tF . By further refining to M3 andM4, we can obtain mesh convergence for maximum vorticity in time.

(a) (b)

Figure 8. Vorticity contours at time (a) t D 1.0 and (b) t D 5.0 with forcing time tF D 0.1.

(a) (b)

Figure 9. Pressure contours at time (a) t D 1.0 and (b) t D 5.0 with forcing time tF D 0.1.



0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1

M1M2M3M4

t

max

Figure 10. Maximum of absolute vorticity in the fluid domain as a function of time t 6 1 with forcing timetF D 0.1.

4.3. Time duration of applied external force

We investigate the effect of the time duration of applied external force on the particle. For time06 t 6 tF , the particle moves downward because of the action of external force F (see Figure 11),then the particle moves freely under the influence of the surface tension. Eventually, the particle

-0.05

-0.04

-0.03

-0.02

-0.01

0

0 0.2 0.4 0.6 0.8 1

t

tF = 0.10tF = 0.07tF = 0.05tF = 0.03tF = 0.02

y

Figure 11. The position of the particle as a function of time t 6 1 for different forcing times tF .

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 10 20 30 40 50

t

tF = 0.10tF = 0.07tF = 0.05tF = 0.03tF = 0.02

y

Figure 12. The position of the particle as a function of time t 6 50 for different forcing times tF .



reaches its equilibrium position in the middle of the channel as shown in Figure 12, as long as theparticle stays in between the fluid–fluid interface during the external disturbance.

4.4. Interfacial thickness

In the diffuse-interface model, the interfacial thickness is defined by � Dp"=˛, as explained in

Section 2. The nondimensional measure for the interface thickness is the Cahn number, defined byC D �=H . Because we scaled every length with channel height H , that is, H D 1, the interfacialthickness � is already nondimensional (� D C ). We change the interfacial thickness � by changing "values while keeping the "DM 2 relationship. Note that C D � D

p"DM because we fix ˛ D 1.

Figure 13 shows the interface of Cahn number 0.01 and 0.04, which means the interfacial thicknessis 1% and 4% of the channel height, respectively. Note that the particle radius is aD 0.15.

Figure 14 shows the histories of the particle position for various Cahn numbers where tF D 0.05with Fy D 10. As the Cahn number increases, that is, the interfacial thickness increases, the particlemoves rapidly towards the equilibrium position.

Here we varied " values to change the interfacial thickness. However, this changes not only theCahn number but also the Capillary number and Péclet number (Equation (13)). Hence, Figure 14is manifested by the combination of these nondimensional parameters. In the Cahn–Hilliard model,it is not trivial to change the Cahn number only without affecting other parameters.

H

(a) (b)

Figure 13. Interfacial thickness for the Cahn number (a) C D 0.01 and (b) C D 0.04.

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 5 10 15 20

t

y

C = 0.01C = 0.02C = 0.03C = 0.04

Figure 14. Effect of the Cahn number on the position of the particle as a function of time t 6 20.



4.5. Surface tension

The nondimensional measure of the surface tension is described by the capillary number Ca D��U=�"c2B , which contains a characteristic velocity U . Because the fluid and particle velocities areunknowns and part of the solution, it is not trivial to define a characteristic velocity U , prior to solv-ing the given problem. In our simulations, we change the value of � to change the surface tension,while fixing the other values � D 0.01, � D 1.0, " D 10�4, and M D 10�2. By increasing �, thesurface tension increases (the capillary number Ca decreases).

Figure 15 shows the histories of particle position for various � values where tF D 0.05 withFy D 10. As � increases, that is, the surface tension increases, the particle moves rapidly towardsthe equilibrium position. The results are quite intuitive; because the driving force pulling the particleback to its original equilibrium position is the surface tension, the particle will return faster underhigher surface tension.

4.6. Particle size

Figure 16 shows the histories of the particle position for various particle radii where tF D 0.05 withFy D 10. As the particle radius a decreases, the particle moves further downward when externalforce F is applied because smaller particle experiences less drag than larger one. After the externalforce is released, smaller particle moves faster than larger one, which can be seen in Figure 17 wherethe y-directional translational velocity of the particle V is shown.

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 5 10 15 20

t

= 100= 200= 300= 400y

Figure 15. Effect of the surface tension on the position of the particle as a function of time t 6 20. As �increases, the surface tension increases.

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 5 10 15 20

t

a = 0.100a = 0.125a = 0.150a = 0.175a = 0.200y

Figure 16. Effect of the particle size on the position of the particle as a function of time t 6 20.



0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

V

a = 0.100a = 0.125a = 0.150a = 0.175a = 0.200

Figure 17. Effect of the particle size on the velocity of the particle as a function of time t 6 1.

4.7. Viscosity ratio

We now consider the effect of different viscosities of upper and lower fluids. We define the viscosityratio � D �2=�1. Figure 18 shows the histories of the particle position for different viscosity ratioswhere tF D 0.05 with Fy D 10. As the viscosity ratio decreases, that is, the upper fluid is thinnerthan the lower fluid, the particle moves further downward when external force F is applied. Afterthe external force is released, the particle moves faster for lower viscosity ratios, eventually theparticle reaches the equilibrium position earlier. The fast movement of the particle for lower viscos-ity ratios is clearly seen in Figure 19, where the y-directional translational velocity of the particleV is plotted after the external force F is released.

5. MULTILAYER CONFIGURATION

In this section, we show the dynamics of a particle passing through multiple layers of fluids,confined between two parallel plates. A schematic description of the problem is shown in Fig-ure 20. The length of the channel is L D 1, the positions of the uppermost and lowest fluids areH1 D H5 D 0.6 and the positions of the fluids in-between are H2 D H3 D H4 D 0.3. The vis-cosity of the fluid layers is chosen � D 1, but the model can handle different viscosity ratios asshown in the previous section. A particle of radius a D 0.15 is suspended at the initial positionX0 D .0, 1.8/ and is sedimenting downward as a result of a constant external force F D .0,�10/acting on the particle. The upper and lower walls are stationary and the flow is assumed to be

-0.025

-0.02

-0.015

-0.01

-0.005

0

0 5 10 15 20

t

= 0.7= 0.9= 1.1= 1.3

y

Figure 18. Effect of the viscosity ratio on the position of the particle as a function of time t 6 20.



0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t

V

= 0.7= 0.9= 1.1= 1.3

Figure 19. Effect of the viscosity ratio on the velocity of the particle as a function of time t 6 1.

x

y

L

H1

H2

H3

H4

H5

F

c = +1

c = +1

c = +1

c = −1

c = −1

Figure 20. Geometry for a particle in multiple layers of fluids.

periodic in the x direction. The material parameters used in governing Equations (2), (4), and (8)are ˛ D 1, ˇ D 1, " D 0.0001, M D 0.01, and � D 100. Note that the interfacial thickness� D

p"=˛ D 0.01.



The problem is solved using a mesh with 100 � 210 elements because it provides an accuratesolution with manageable computational costs. The time step t D 0.001 is used for the simula-tion. Figure 21 shows the evolution of the structure of the multilayered morphology caused by thesedimenting particle. As the particle passes through the multiple layers of fluids, a fluid layer breaks

(a) t = 1.5 (b) t = 3.0 (c) t = 4.0

(d) t = 5.0 (e) t = 5.5 (f) t = 6.0

Figure 21. Snapshots of a particle passing through multiple layers of fluids.

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0 1 2 3 4 5 6 7

t

V

Figure 22. Translational velocity of the particle passing through multiple layers of fluids as a function of time.



up into several drops, then the drops merge with other layers of the fluid. Figure 22 shows the trans-lational velocity of the particle in the y direction as a function of time. The proposed method canprovide fully resolved velocity fields for the particle and fluids, associated with the evolution of theinherently complex morphology.

6. CONCLUSIONS

We presented a combined model of the Cahn–Hilliard theory and XFEM for the dynamics of parti-cles suspended in two-phase flows. In the diffuse-interface model of Cahn–Hilliard, the interface isconsidered to have a small but finite thickness. The interface profile and thickness are determinedby governing equations that couple thermodynamic and hydrodynamic forces.

For the direct numerical simulation of flows with suspended particles, we use XFEM, whichdecouples the fluid and particle domains while using a computational mesh including both fluidsand particles. To cope with the movement of particles, a tALE scheme is used to define a mappingof field variables at previous time levels onto the computational mesh at the current time level. Theno-slip boundary condition at the particle surface is imposed by using a constraint implemented withLagrangian multipliers. By combining the diffuse interface model and XFEM, the particle motionat a fluid–fluid interface can be simulated on a fixed Eulerian mesh without any need of re-meshing.

We present the motion of a single particle at an interface between two fluids. Initially, the fluidsand particle are stationary. The initial equilibrium state is disturbed by applying a constant forceon the particle for a certain time duration. Then the external force on the particle is released, andthe particle moves freely to its equilibrium position under the action of surface tension. As long asthe particle stays in between two fluids during external disturbance, it always comes back to its ini-tial equilibrium position. We investigated the effect of interfacial thickness, surface tension, particlesize, and viscosity ratio of the two fluids on the particle’s movement towards its equilibrium posi-tion. As interfacial thickness increases, surface tension increases, particle size decreases or viscosityratio decreases; the particle moves rapidly towards its equilibrium position after the external force isreleased. We also demonstrated the wide applicability of the method and determined the motion ofa sedimenting particle passing through multiple layers of fluids and the corresponding morphologychange of the fluids. The proposed method is general and is applicable to more complex problems,such as multiple particles in phase separating fluids and structure formation of particles at a fluid–fluid interface. Also, the method can be easily extended to three-dimensional simulations withoutany loss of generality, only requiring heavier computational load. Future work will be focused onthese problems.

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