multiphase lattice boltzmann method for particle...

Multiphase lattice Boltzmann method for particle suspensions

Abhijit S. Joshi and Ying Sun*State University of New York at Binghamton, Binghamton, New York 13902, USA

Received 3 December 2008; published 4 June 2009

A two-dimensional mass conserving lattice Boltzmann method LBM has been developed for multiphaseliquid and vapor flows with solid particles suspended within the liquid and/or vapor phases. The mainmodification to previous single-phase particle suspension models is the addition of surface adhesive forcesbetween the suspended particle and the surrounding fluid. The multiphase dynamics between fluid phases issimulated via the single-component multiphase model of Shan and Chen Phys. Rev. E 47, 1815 1993. Thecombined multiphase particle suspension model is first validated and then used to simulate the dynamics of asingle-suspended particle on a planar liquid-vapor interface and the interaction between a single particle and afree-standing liquid drop. It is observed that the dynamics of suspended particles near free-standing liquiddroplets is affected by spurious velocity currents although the liquid-vapor interface itself is a local energyminimum for particles. Finally, results are presented for capillary interactions between two suspended particleson a liquid-vapor interface subjected to different external forces and for spinodal decomposition of a liquid-vapor mixture in the presence of suspended particles. Qualitative agreements are reached when compared withresults of suspended particles in a binary mixture based on multicomponent LBM models.

DOI: 10.1103/PhysRevE.79.066703 PACS numbers: 47.11.j, 05.20.Dd, 68.03.g, 68.08.Bc

I. INTRODUCTION

Using the lattice Boltzmann method LBM to study sus-pensions of solid particles in a single-phase fluid was pio-neered by Ladd 1,2. Over the years, it has attracted greatinterest within the LBM research community and severalversions of the original algorithm are now available 3–8.The LBM model for suspensions has been proven to be apromising simulation tool in applications ranging from col-loidal suspensions 9,10 to biofluids 11 and shows verygood scaling on large-scale parallel computing platformswith up to a million particles 12. On another front, therehas been a rapid development of multiphase and multicom-ponent LBM models 13–18. However, combining thesetwo research endeavors has not been pursued until very re-cently 19–21. These recent efforts have focused on the mo-tion of suspended particles in the interface or bulk regions ofmulticomponent fluids, having similar densities.

In this paper, we report some results of combining theShan and Chen single-component multiphase SCMP modelwith a model for particle suspensions for a system composedof liquid and vapor, with suspended particles in one or bothof the fluid phases. A unique challenge of this approach is toaccurately model the coexistence of at least three phases liq-uid, vapor, fixed solid walls, and/or moving suspended par-ticles. Unlike the multicomponent LBM, the multiphaseLBM involves a large density ratio between fluid liquidand vapor phases and b less flexibility in independentlyvarying the interface tensions between the liquid, vapor, andsolid phases due to the strong correlation between the inter-face tension and cohesive/adhesive forces. Modificationsmade in the LBM algorithm to make the model mass con-serving and stable are the following:

i Treatment of interior fluid nodes inside the suspendedparticles. Like the original motivation given by Ladd 1,2,

the interior fluid inside particles is maintained to avoid cre-ating and destroying fluid at nodes as the particle moves onthe lattice. Thus, unlike methods where the internal fluid iscompletely removed 3, the problem of initializing newlycreated fluid nodes originally inside the moving particledoes not arise and the mass of the entire system is exactlyconserved. The collision, streaming, and bounce-back stepsin the LBM algorithm described in detail in Secs. II and IIIare carried out even in this interior fluid, with some minorchanges that will be described later. However, the velocity atinterior fluid nodes is always reset to the particle velocityassuming that the entire fluid phase inside a particle moveslike a rigid body with the same velocity as that of the particleitself. Collisions of the interior fluid with the particle bound-ary do not contribute directly to the force calculations on thesuspended particle, similar to the virtual fluid used in Dingand Aidun 8. However, it influences the force calculationindirectly when the particle representation on the LBM lat-tice changes. Finally, following Heemels et al. 7, we elimi-nate the effect of sound propagation in the interior fluid byaveraging and redistributing the density.

ii Inclusion of adhesive forces acting on the suspendedparticles. Adhesive forces between the external fluid outsidesuspended particles on the suspended particle are introducedsuch that they are equal and opposite to the adhesive forcesexerted by the solid on the fluid. These forces are not presentfor the internal fluid inside particles. These forces can beused to control the wetting properties of suspended particlesrelative to the liquid-vapor interface.

The subsequent part of the paper is organized as follows.In Sec. II, the Shan and Chen multiphase model is summa-rized. Section III gives a detailed discussion of the particlesuspension model. Section IV describes several validationcases for individual submodels. The combined multiphase+particle suspension model is then used in Sec. V to simu-late various problems where the suspended particles are con-fined within the liquid/vapor or are present in both phases.The model and results are summarized in Sec. VI. Envi-*[email protected]

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sioned applications of the model are to predict the wetting,dewetting, contact line pinning, and particle self-assembly ofa drop containing colloidal particles as it spreads on varioustypes of surface energy heterogeneous and/or rough sub-strates.

II. MULTIPHASE LATTICE BOLTZMANN METHOD

The LBM can be viewed as a discrete approximation ofthe Boltzmann equation 22. But in most implementations,the LBM is an alternative computational fluid-dynamics ap-proach for solving the Navier-Stokes equations, which canbe derived from the LBM using the Chapman-Enskog pro-cedure 23 or an asymptotic expansion technique 24. Thissection is a brief summary of the isothermal two-phaseliquid+vapor LBM model introduced by Shan and Chen14. The LBM simulates fluid flow on a discrete lattice withequally spaced nodes along the x and y directions repre-sented by square symbols in Fig. 1a. The primary depen-dent variables at each node are the particle velocity distribu-tion functions PDFs along different lattice directions .The PDF along a certain direction represents the number offluid particles moving in that direction at that time and isindicated by f. The transport of these PDFs along discretelattice directions and their interactions via suitably designedcollision terms can reproduce the dynamics of the Navier-Stokes equations. At any lattice node x, the evolution of thePDF with time t is governed by the lattice Boltzmann equa-tion given by

fx + e,t + 1 = fx,t − fx,t − feqx,t,ux,t

+ feqx,t,ux,t − f

eqx,t,ux,t .

1

The right-hand side in Eq. 1 represents the collision stepand equating the right-hand side to the left-hand side repre-sents the streaming step. The relaxation time in Eq. 1 con-trols the kinematic viscosity of the lattice Boltzmann fluidvia the relationship = 2−1 /6. All equations in this workare presented in lattice units, where the lattice spacing alongthe x and y axes and the time step are all unity x=y=t=1. The LBM results can be compared with experi-ments or with other theoretical/numerical results via the cal-culation of suitable dimensionless numbers. The fundamentalprinciple used is that dimensionless numbers are the same inlattice units and physical units. Examples of such compari-sons are given later in Secs. IV A and IV C. Note that the lasttwo terms in Eq. 1 are the result of using the exact differ-ence method EDM introduced by Kupershtokh andMedvedev 25. The EDM ensures that the density ratio be-tween the liquid and vapor phases is not affected when therelaxation time is different than unity or when differentvalues of are used in the liquid and vapor phases.

The discrete velocities e depend on the particular veloc-ity model used and we use the D2Q9 model that has ninevelocity directions =0 to 8 at a given lattice point, withthe individual components along the x and y directions givenby

e0 = 0,0, e1 = 1,0, e2 = 1,1 ,

e3 = 0,1, e4 = − 1,1, e5 = − 1,0 ,

e6 = − 1,− 1, e7 = 0,− 1, e8 = 1,− 1 . 2

The macroscopic density and velocity u at each fluidnode are obtained using

= =0

8

f, 3

u = =1

8

fe. 4

These macroscopic quantities are used to evaluate the equi-librium PDF f

eq given by

feq,u = w1 + 3e · u +

9

2e · u2 −

3

2u · u2 .

5

The weight functions for different directions are w0=4 /9,w=1 /9 for =1,3 ,5 ,7 and w=1 /36 for =2,4 ,6 ,8. Themodified velocity u to account for particle interactions, asintroduced in Shan and Chen 14 and modified by Kupersh-tokh and Medvedev 25, is given by

u = u +1

Fcohesive + Fadhesive + Fbody . 6

The modified velocity u from Eq. 6 replaces u in Eq. 4while calculating the equilibrium function in the third termon the right-hand side of Eq. 1. Finally, the actual fluidvelocity u used for plotting velocity vector fields is theaverage velocity before and after the collision event and isgiven by

R

U

U

U

C

C

C

FIG. 1. a Mapping of a circular particle of radius R dashedline on the LBM lattice, showing external fluid nodes whitesquares and internal fluid nodes dark squares. The shaded regionis the effective shape of the particle. The fluid interacts with theparticle at the boundary nodes black circles. b Change in theparticle representation on the lattice during one lattice time step,with the shaded circle representing the previous location and thedashed circle representing the new location. Some internal fluidnodes are uncovered U and become external fluid nodes, whilesome external fluid nodes are covered C by the particle and be-come internal fluid nodes.

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u = u +1

2Fcohesive + Fadhesive + Fbody . 7

The cohesive force at location x arises because of theattraction between fluid particles at x and fluid particles atneighboring locations x+e. This attraction is proportional tothe effective density 14, x=x=1−exp−, at in-teracting nodes and is calculated using

Fcohesivex = − =1

8

Gfluid-fluidxx + ee. 8

For the D2Q9 model, Gfluid-fluid=gf for =1,3 ,5 ,7 and

Gfluid-fluid=gf /4 for =2,4 ,6 ,8. If the interparticle cohesive

force parameter gf is negative, attractive forces are set upbetween fluid at neighboring nodes on the lattice and thisleads to the formation of a high-density liquid phase sur-rounded by a low-density vapor phase. Larger values of gfcorrespond to greater attractive forces, leading to higher-density ratios between the liquid and vapor phases. Caremust be taken while using the expression in Eq. 8 nearsolid walls or near suspended particles. In such a case, weuse a wall density value w=c=0.693 for the solid nodeor the fluid node inside the suspended particle and set =w to calculate the effective density at the solid node26. Thus, an adhesive force is introduced between the fluidand solid nodes independent of the adhesive force discussednext. The wetting behavior of a liquid drop in contact with asolid wall can be controlled entirely by adjusting this walldensity. We find that using w=0.693 ensures that the contactangle is close to 90° when the adhesive force parameter gw tobe discussed later is set to zero. The pressure P can be ob-tained using the equation of state EOS. For the D2Q9model, the EOS is given by 25

P =

3+

3

2gf

2. 9

The speed of sound cs=1 /3 and the critical density is c=0.693. Phase separation occurs when gf −4 /9. The equa-tion of state can be changed by modifying the expression foran effective density 27.

We control the degree of departure of the contact anglefrom 90° via an additional adhesive force given by

Fadhesivex = − =1

8

Gfluid-solidxx + ee. 10

For the D2Q9 model, Gfluid-solid=gw for =1,3 ,5 ,7 and

Gfluid-solid=gw /4 for =2,4 ,6 ,8. The function sx+e=1

if the location x+e corresponds to a solid node or a fluidnode inside suspended particles and sx+e=0 otherwise.Thus, the adhesive force is present only if one or more of theneighbors of a fluid node at location x is solid. The magni-tude of the adhesive force is controlled by the parameter gw,with positive values of gw leading to repulsive forces non-wetting behavior and negative values of gw leading to attrac-tive forces wetting behavior. Note that the fluid sticks tothe wall even when gw=0 because of the cohesive attraction

between the solid node and the adjacent fluid node, as dis-cussed in the previous paragraph.

The interaction force-based multiphase model leads to thedevelopment of spurious velocity currents based on Eq. 7close to the liquid-vapor interface, when the interface curva-ture is nonzero. In general, the magnitude of these currentsusmax increases with the density ratio L /V between theliquid and vapor phases. Our computations for liquid dropsshow that usmax=0.0046 for L /V=10 and usmax=0.0265for L /V=30. The presence of these currents can signifi-cantly influence the motion of suspended particles close tothe interface, as shown later in Sec. V B. For a given densityratio, these spurious currents can also be reduced by incor-porating additional rings of neighboring nodes while calcu-lating adhesive and cohesive interactions 28.

III. PARTICLE SUSPENSION MODEL

For simplicity, the suspended particles hereafter referredto as simply “particles” are circular in shape and the loca-tion of each particle, the particle radii, and the linear andangular velocities are specified as part of the initial condi-tions. To allow the particle to interact with the LBM, nodesinside and outside the particle are identified as shown in Fig.1a. Boundary nodes are located half way between fluidnodes inside the particle and external fluid nodes. As theparticle moves, this mapping is updated. Initially, nodes ly-ing in the liquid phase are assigned the liquid density andnodes in the vapor phase are assigned the vapor density,corresponding to the value of the cohesive force parametergf. Note that the fluid phase is present at all nodes in thedomain, except fixed solid nodes, if any. The fluid velocity isinitially set to zero at all nodes unless otherwise specified.After initialization, each time step involves the followingsubsteps:

1 Updating particle positions based on the forces andtorques acting on each particle.

2 Calculation of forces and torques due to the change inparticle representation on the lattice.

3 Calculation of equilibrium distributions.4 Streaming includes bounce back at fixed walls and at

moving particle interfaces.5 Updating the force and torque on each particle.6 Updating macroscopic variables density and velocity

in the fluid phase.In the following, each of the above steps is presented in

detail.

A. Particle dynamics and kinematics

This step is made up of two substeps. The first substep isdynamics, where Newton’s second law of motion is appliedto each individual particle to calculate the changes in thelinear and angular velocities Up and p, respectively be-cause of the total force Fp and total torque Tp acting on it,using

Fp = Mpd

dtUp, 11

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Tp = Ipd

dtp. 12

For an explicit update, the linear velocity at the new timelevel n+1 is obtained using forces calculated at the previ-ous time level n and the discrete version of Eq. 11 isgiven by Up

n+1=Upn +Fp,n /Mp. A similar expression can be

used to update the angular velocity. However, an explicitupdate of the linear and angular velocities especially thelatter based on Eqs. 11 and 12 is unstable for low particledensities. This problem can be avoided by using an implicitupdate 6 Appendix. For the explicit update, we followLadd 1 and use time averaging over two successive timesteps to calculate the forces. We assign a solid density to theparticles and use the following relationships to determine themass and moment of inertia for a particle of density p andradius R:

Mp = pR2, 13

Ip =1

2pR2. 14

The total force Fp in Eq. 11 is composed of the forces dueto the changing representation of the particle on the latticeFC

p and FUp , force due to the momentum transfer from the

fluid Fbbkp , adhesive force Fadh

p , and lubrication and Hookeanforces acting between pairs of particles denoted by Flub

p andFHook

p , respectively superscript p denotes that the force isacting on the particle. Thus,

Fp = FCp + FU

p + Fbbkp + Fadh

p + Flubp + FHook

p . 15

Similarly, the total torque is obtained using

Tp = TCp + TU

p + Tbbkp + Tadh

p . 16

Lubrication and Hookean forces are assumed to act along aline connecting the centers of mass of the involved particlepairs and therefore do not contribute to the torque. The cal-culation of all these forces and torques is discussed in Secs.III B and III E.

The next substep is kinematics, where the updated linearand angular velocities are used to calculate the new positionand angular orientation of each particle, the latter being re-dundant if the particles are circular. Note that although par-ticle positions and velocities are updated continuously intime as their dynamics evolves, the representation of the par-ticle on the discrete LBM lattice may not change at everytime step.

B. Force and torque due to changing particle representationon the lattice

When the particle moves from one time level to the next,it may cover some nodes that were formerly external fluidnodes. These nodes are indicated in Fig. 1b by the letter Cand their location at the new time level is denoted by rC. Thecenter of mass of the particle at the new time level is denotedby R. In such a case, the momentum at these fluid nodes istransferred to the particle and the resultant force and torqueon the particle can be calculated using

FCp =

C

CuC 17

and

TCp =

C

rC − R CuC. 18

The density and velocity values from the previous time stepwhen node C was an external fluid node are used in Eqs.17 and 18.

Similarly, some nodes that were originally inside the par-ticle can be uncovered and become external fluid nodes.These are indicated in Fig. 1b by the letter U and are lo-cated at rU at the old time level. The center of mass of theparticle at the old time level is R. In this case, the momen-tum corresponding to the fluid motion is lost by the particleand the resultant force and torque on the particle are obtainedusing

FUp = −

U

UuU 19

and

TUp = −

U

rU − R UuU. 20

The density U represents the average density of the fluidinside the particle at the previous time step when the node Uwas part of the interior fluid and the velocity corresponds tothe rigid-body motion of the particle at the previous timelevel, as discussed later in Sec. III F. Unlike schemes that donot use the interior fluid at all 3, no extrapolations arerequired in the present model to assign density and velocityto nodes uncovered by the moving particle and mass is there-fore exactly conserved. If the particle representation on thelattice does not change from one time level to the next, theforces and torques in this section are zero.

C. Calculation of equilibrium distributions

The density and velocity at each node point includingfluid nodes inside the suspended particles are used to calcu-late the equilibrium distributions given in Eq. 5. As dis-cussed in Sec. III F, the fluid velocity inside each particle isreset to correspond to the rigid-body motion of the particle asa whole. This modified velocity is used to calculate equilib-rium distributions for fluid nodes inside the particle. Thedensity of fluid inside the particle is reset to the averagevalue. This average density is used to calculate equilibriumfunctions for fluid nodes inside suspended particles. In thefluid outside the suspended particle, the velocity used in theequilibrium function is given by Eqs. 4 and 6. Careshould be taken to include the cohesive and adhesive forceterms while calculating the modified velocity at external


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fluid nodes adjacent to the suspended particle, similar to thecase where fluid nodes are adjacent to a fixed solid wall.

D. Streaming and bounce back at fixed and moving solidboundaries

The postcollision PDF f+ at lattice node x along direction

is the right-hand side of Eq. 1 and is defined as

f+ = fx,t − fx,t − f

eqx,t,ux,t

+ f

eqx,t,ux,t − feqx,t,ux,t . 21

It is f+ that will now stream to the adjacent lattice point x

+e. The streaming step from a fluid node outside the par-ticle to a neighboring fluid node also outside the particlecan be described as

fx + e,t + 1 = f+ . 22

Physically, this process can be thought of as a movement offluid particles from location x to location x+e. If the adja-cent node corresponds to a fixed solid node, the half-waybounce-back scheme is used to simulate the no-slip boundarycondition. The streaming step at these nodes can be de-scribed as

f x,t + 1 = f+ , 23

where the direction is defined to be the direction oppositeto . The no-slip boundary is effectively present at a locationmidway between the fluid node and the fixed solid node.

Finally, we consider the case where streaming occursfrom a fluid node just outside a moving particle toward anode that is inside the particle. Based on the known particlevelocity, we first determine the velocity of the boundarynode assumed to be located midway between the fluid nodeand the node inside the particle and indicated by the filledcircle in Fig. 1a. The boundary node velocity includes bothtranslational and rotational components of the particle veloc-ity and is given by

ub = U + rb − R , 24

where U is the translational velocity of the particle, is theangular velocity, rb is the location of the boundary node, andR is the location of the center of mass of the particle. Thestreaming step can now be described as

f x,t + 1 = f+ − 6wwub · e . 25

Note that if the boundary node velocity is zero, Eq. 25reduces to the conventional bounce-back rule. There is asmall change in mass associated with the modified bounce-back rule in Eq. 25 because of the second term on theright-hand side. This can be exactly compensated by using asimilar bounce-back rule for the fluid inside the suspendedparticle and using the same wall density for both cases. Inthis work, we set w to be the density at the external fluidnode.

If the surfaces of two particles are closer than one latticespacing along a certain link, there are no external fluid

nodes separating them along that link. To ensure mass con-servation, streaming along this link from the interior fluid ineach particle is carried out assuming that the neighboringfluid node inside the other particle is a fixed solid node. Itmust be mentioned that accurate LBM schemes have beendevised to implement the no-slip boundary for curved mov-ing boundaries 29–33. However, most of these schemessuffer from an inability to conserve mass because of the in-terpolations used. We have chosen to use the half-waybounce back rule both for simplicity and in order to maintaina mass imbalance close to machine zero.

E. Updating the force and torque on each particle

At each boundary node point x in the external fluid alongthe direction e, where x+e is an internal fluid node, ahydrodynamic force is exerted because of the change in mo-mentum of the fluid particles impinging along e repre-sented by f+ and getting reflected back along e representedby f

+ −6wwub ·e. The total force and torque on the par-ticle are calculated by summing these interactions along allboundary nodes bn and along all relevant directions foreach boundary node using

Fbbkp =

bn

2f+ − 6wwub · ee 26

and

Tbbkp =

bn

rbn − R

2f+ − 6wwub · ee. 27

Similarly, the total adhesive force acting on the particlecan be obtained by adding up the adhesive force contribu-tions at all boundary nodes and along all relevant directionsat each boundary node. The adhesive force and torque actingon the particle is given by

Fadhp =

bn

Gfluid-fluidxx + ee

+ Gfluid-solidxx + ee 28

and

Tadhp =

bn

rbn − R

Gfluid-fluidxx + ee

+ Gfluid-solidxx + ee . 29

The density values used in Eqs. 28 and 29 are exactlyidentical to those used while calculating the adhesive andcohesive forces on the external fluid because of the particle.

If the distance h between the surfaces of two suspendedparticles 1 and 2 of radii a1 and a2 is small, there areinsufficient external fluid nodes between the moving sur-faces to resolve the lubrication forces. To correctly modelthis effect in the LBM, additional forces must be introducedbetween each such pair of particles when their respectivesurfaces get closer than a certain cut-off distance hc. For thetwo-dimensional 2D case, we use the expression derived byKromkamp et al. 10 to model the lubrication force betweena pair of interacting cylinders. The force on particle 1 isgiven by


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Flubp = 0,h hc,

Flubp = −

1

2 U12 · R12a1 + a2

h 3/2F0 +

h

a1 + a2F1

− a1 + a2

hc 3/2F0 +

hc

a1 + a2F1 , h hc. 30

In Eq. 30, R12 is the unit vector from the center of particle1 toward particle 2, U12 is the velocity of particle 2 relativeto particle 1, is the dynamic viscosity, and the constantsare F0=32 /4 and F1=2312 /80. The critical spacingused is hc=2. Note that Eq. 30 is applied pairwise to allparticles whose outer surfaces are within the cut-off distancefrom each other.

In addition to the lubrication force, we introduce aHookean repulsion between particles, identical to that usedin Kromkamp et al. 10 if their respective surfaces comewithin 0.1 lattice spacing of each other. This can sometimeshelp prevent the breakdown of the calculations when the par-ticle density is high and there is considerable clustering to-gether of the particles. This Hookean force on particle 1 isobtained using

FHookp = 0,h ,

FHookp = − F01 − h/R12,h . 31

The constant F0 can be adjusted to control the magnitude ofthe repulsive force and the cut-off spacing =0.1. We use avalue of F0=10. In actual colloidal suspensions, additionalforces between colloidal particles are present and these canbe added in the model if required.

F. Calculation of macroscopic variables (density and velocity)

At external fluid nodes outside the particles, this step isessentially similar to that described by Eqs. 3 and 4.However, some modifications are introduced for the fluidinside the particles. The density is still calculated via Eq. 3,but the velocity at each node inside the particle is assigned avalue equal to the particle velocity, similar in form to Eq.24. Thus, the fluid inside the particle is forced to assume arigid-body motion corresponding to the particle. In addition,to remove the effects of the sound propagation in the interiorfluid inside each particle, the average density inside eachparticle is calculated and the density at each node inside thatparticle is reset to the average density, as in Ref. 7. Thisoperation conserves mass and eliminates density fluctuationsin the interior fluid. Because of the interior fluid, the densityand velocity are not recalculated while evaluating forces dueto the changing representation of the particle on the latticeSec. III B.

The calculations described in Sec. III A to Sec. III F arerepeated until the system reaches a steady state or until thedesired transient behavior is simulated. For two-phase prob-lems where suspended particles are introduced near liquiddrops, the liquid phase is allowed to reach equilibrium withthe vapor phase before particle dynamics begins. This initial

equilibration can take several thousand time steps but may beimportant to reduce effects of the initial condition on thesubsequent particle dynamics 34.

IV. MODEL VALIDATION

A. Single-phase model without suspended particles

The exact difference method 25 used in the presentLBM for incorporating external forces was validated againstthe exact solution to the Navier-Stokes equations for Poi-seuille flow in an infinitely long 2D channel. In this problem,the top and bottom walls of a channel at y=0 and y=H are atrest and the flow is driven by a constant body force Fbody=gxi along the x direction. The fluid density and dynamicviscosity are and , respectively. At steady state, the flowprofile is parabolic, with the maximum velocity at the cen-terline of the cavity given by Umax=gxH

2 / 8 . Defining thenormalized velocity u=u /Umax and the dimensionless dis-tance from the bottom plate y=y /H, the steady-state veloc-ity profile is given by u=4y1−y.

We simulate this problem in the LBM using a 11105lattice with periodic boundaries along the x direction andfixed solid nodes at j=1, j=2 and j=103, and j=104 repre-senting the bottom and top walls. Based on the half-waybounce back rule used, the channel height H=100. Otherparameters in the LBM in lattice units are gx=1 /100000,=2.54, and =0.9. Based on the relaxation time , the ki-nematic viscosity = 2−1 /6=0.1333. The dynamic vis-cosity ==0.3386 and Umax=0.0369. The steady-state ve-locity profile obtained using the LBM is first normalizedusing H=100 and Umax=0.0369 to make it dimensionlessand then compared with the exact solution given above. Fig-ure 2 shows that an excellent agreement between the LBMand the exact solution is obtained. The inset in Fig. 2 shows

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.

u*

y*

Exact Solution

Present LBM

j = 1

j = 2

solid wall

FIG. 2. Validation of the LBM with an exact solution to theNavier-Stokes equations for the case of body-force-driven Poi-seuille flow in an infinite 2D channel. The velocity is normalized bythe maximum velocity and the distance is normalized using thechannel height. The inset shows the placement of lattice nodes andthe velocity profile relative to the fixed wall y=0 at the bottom ofthe channel.


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the placement of the lattice nodes near the bottom wall. Notethat there is no node at the precise location of the wall, butthe no-slip boundary condition is enforced between the lat-tice nodes on either side of the wall between j=2 and j=3.The LBM simulations were repeated for relaxation times inthe range 0.61.5, and an excellent agreement with theanalytical solution was obtained for all cases. The only con-straint is that the LBM velocity in lattice units should besmall compared to the speed of sound cs=1 /3 in latticeunits. This ensures that the simulations take place in theincompressible flow regime. In practice, accurate solutionsfor the incompressible flow can be obtained for 0Umax /cs0.1, beyond which the LBM can become un-stable.

B. Multiphase model without suspended particles

The SCMP model was used to study the phase-separationprocess of an initially single-phase fluid into distinct liquidand vapor phases. The driving mechanism of this phase sepa-ration is the cohesive force between fluid molecules locatedat adjacent lattice nodes. The simulation was carried out fora fluid on a periodic domain with an initially uniform densityof 0.693 corresponding to c. At time t=0, a small0.5% perturbation is introduced in the density. As ex-pected from the equation of state, if the cohesive force is lessthan a critical value gf −4 /9, these density perturbationsgrow with time and successively coarser liquid structures areformed throughout the domain. The density ratio between theliquid and vapor phases and the interface tension liq-vap be-tween the two phases are controlled by the cohesive forceparameter gf. These simulations are useful in tabulating equi-librium liquid and vapor densities for various cohesive fac-tors. The equilibrium densities can be used as initial condi-tions for subsequent simulations that start directly with afinite-sized liquid drop surrounded by the vapor phase. In theoriginal Shan and Chen model 14, the equilibrium liquidand vapor densities change with the relaxation time even ifgf is constant. The effect of on the density ratio is elimi-nated by using the exact difference method 25. To calculatethe interface tension liq-vap, we measure the pressure differ-ence P inside and outside a liquid drop of radius R inequilibrium with vapor for different drop sizes. This pressuredifference is related to the interface curvature via the Laplaceequation

P =liq-vap

R. 32

Depending on the radius of the drop, the pressure field takesup to 50000 lattice time steps or more to stop oscillating andreach steady state. Pressure measurements are taken after asteady pressure profile is obtained along the line through thedrop center indicated by AB in Fig. 3a. There are largefluctuations in the pressure distribution near the liquid-vaporinterface. These fluctuations are an artifact caused by thesharp increase in density and are ignored while calculatingthe pressure difference. A typical example of the density andpressure distributions for R=75 is also shown in Fig. 3ainsets. At equilibrium, the average pressure inside the liq-

uid drop away from the interface is slightly larger than thepressure outside it by an amount P. Based on Eq. 32, theslope of the line in a plot of P versus the curvature of theliquid drop, as shown in Fig. 3b, gives the interface ten-sion. For the simulations in Fig. 3, gf =−0.65 this leads toliq=2.53 and vap=0.08 and liq-vap=0.13.

Next, using the SCMP model, we calculate the equilib-rium contact angle for a liquid drop on a perfectly flat anduniform substrate inset in Fig. 4a. The contact angle isrelated to the liquid-solid interface tension liq-sol, the vapor-solid interface tension vap-sol, and the liquid-vapor interfacetension liq-vap via Young’s law,

cos =liq-sol − vap-sol

liq-vap . 33

In the LBM, the contact angle can be changed by varying theadhesive force parameter gw. Note that the density ratio be-

A

B

Pressure ( P ) along line AB

X

∆P

Pressure

Density

0.0240

50 100 150 200 250 300

0.0235

0.0230

0.0225

0.0220

0.0215

0.000

0.001

0.002

0.003

0.004

0.005

0.00

0.00

0 0.02 0.04 0.0

B

alaes la

1

∆P

σ 0.13

FIG. 3. Color online Using Laplace’s law to obtain the inter-face tension in the LBM model using a 300300 periodic latticewith gf =−0.65. a Pressure and density distribution for a liquiddrop with radius R=75. The density ratio is 30. b Summary ofLBM results for drops of different radii and calibration of the inter-face tension liq-vap using Laplace’s law. The solid line is drawnassuming liq-vap=0.13.


066703-7

tween the liquid and vapor phase is still controlled by thecohesive force parameter, which is set to gf =−0.65. The 2Dresults using the present LBM and similar results for a three-dimensional 3D LBM implementation by Yuan 35 arecompared in Fig. 4a. The trend of the results is comparable,but some deviations are observed, especially for gw0. Thecontact angle plug in from IMAGEJ 36 was used for mea-suring contact angles. The deviations may be due to differ-ences in the wall density contribution to the adhesive forceand in the density ratio used, the details of which are notfully disclosed by Yuan 35. A direct comparison of theLBM results with Eq. 33 cannot be made because of thedifficulty in obtaining the interface tension values liq-sol andvap-sol directly from the adhesion parameters. Huang et al.37 recently proposed an empirical scheme for determiningcontact angles directly from the LBM adhesion parametersfor a multicomponent system. Their results are not directlyapplicable to the current multiphase system; but on a similarnote, we propose the following empirical relationship for thecurrent multiphase system:

cos =gwliq − gwvap

gfliqvap. 34

In Fig. 4b, we plot the left-hand and right-hand sides of Eq.34 to show that the empirical correlation introduced abovegives results that are reasonably accurate. Thus, estimates ofthe relevant interface tension values can be obtained by com-paring Eq. 34 with Eq. 33. We note that the liquid-vaporinterface tension obtained in Fig. 3b for the same set ofparameters is close to the denominator in Eq. 34, indicatingthat the order of magnitude of the predictions for the liquid-solid and vapor-solid interface tensions in the numerator ofEq. 33 will be of comparable accuracy. A more thoroughanalysis of this empirical relationship or some alternativeexpressions to predict the contact angle in a multiphase LBMwill be useful but are beyond the scope of this work.

C. Particle suspension model in a single-phase fluid

To validate the LBM model of the force exerted on amoving particle, we have carried out a series of simulationsfor calculating the drag force FD exerted on a nonrotatingcircular cylinder of diameter D moving with a uniform ve-locity U through a single-phase fluid of density . The fluidis bounded by two parallel walls at the top and bottom andthe moving cylinder is located exactly in the middle of thesewalls inset in Fig. 5a. The results have been quantified byplotting the drag coefficient CD=2FD / U2D as a functionof the Reynolds number based on the particle diameter Re=UD /. The Reynolds number was varied by changing thevelocity of the suspended particle along the x axis from0.00375 to 0.12. The diameter of the moving particle wasD=25 and the width of the channel in the vertical directionwas L=100. The effect on the drag coefficient of periodicboundaries along the direction of motion of the particle canbe reduced by increasing the lattice size along that direction.We used lattice sizes ranging from 1200100 for low Re to3000100 for higher Re. In Fig. 5a, the results for thedrag coefficient from the LBM are compared with the resultsof Feng et al. 38 for L /D=4 and the LBM predictions areclose to the published values for small Re but seem to devi-ate for large Re. The case L /D=4 was also solved via themore accurate bounce back scheme for curved boundaries29, using the linear interpolation. This improved the LBMpredictions compared to the half-way bounce back rule.Simulations were also carried out for L /D=30 and the LBMresults were compared with the data of Sucker and Brauer39. It can be seen from Fig. 5a that the agreement is goodfor low Re but not so good for high Re. This discrepancy atlarge Re is mainly because of the effect of periodic bound-aries and can be reduced by using a larger lattice size alongthe cylinder motion. Note that the motion of the cylinder isexternally constrained to maintain a strictly uniform velocityand the force exerted by the fluid is decoupled from thesuspended particle dynamics. To validate the calculation ofthe torque exerted by the fluid on a rotating particle, we

wg

Contatangle(θ

)

Adhesive fore arameter ( )

θIQUID

VAPO

0

20

40

0

80

100

120

140

10

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Present B

Yuan (2005)

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

os(θ)

vaporliquidf

vaporwliquidw

g

gg

ρρρψρψ )()( −

FIG. 4. Color online LBM simulation of different contact angles. a Comparison of the present LBM results with the 3D results ofYuan 35. b Correlation of LBM results based on Eq. 34 for approximating interfacial tension parameters for use in the Young-Duprérelationship. The data points denote LBM results and the solid line is assuming that Eq. 34 holds exactly.


066703-8

carried out a series of LBM calculations using a 300300lattice for a particle of radius R1=100 located at 150, 150and enclosed within a fixed concentric cylindrical enclosureof radius R2=110 see inset in Fig. 5b. The annular gap inbetween was filled with a single-phase fluid of density 0.693.The particle was then rotated about its center at a constantangular velocity and the torque T exerted by the fluid onthe rotating particle was measured. Based on the solution ofthe Navier-Stokes equations for viscous flow between con-centric cylinders, an exact analytical solution for the torqueexerted on the cylinder is given by Donnelly and LaMar 40as T=4 R1

2R22 / R2

2−R12. The LBM predictions have been

compared with the exact solution by plotting the rotationaldrag coefficient CD=2T / R122R12 as a function ofthe gap Reynolds number Re=R1R2−R1 / and the resultsare shown in Fig. 5b. The LBM prediction of the torqueacting on the rotating cylinder matches closely with the exactanalytical result.

D. Coupled particle and fluid dynamics

Thus far, although the particle moved or rotated in thefluid, the force or torque exerted by the fluid on the particleverified to be accurately calculated in the previous sectionswas not allowed to play a role on the dynamics of the par-ticle via Eqs. 11 and 12. This restriction is now relaxed.We validate the two-way dynamic coupling scheme usingtwo simple problems. In the first problem, the particle isinitialized in the liquid phase and is given an initial angularvelocity. The resultant torque exerted by the liquid is allowedto influence the rotational dynamics, as described by Eq.12. Because the particle is circular, the rotation does notchange the representation in the particle on the lattice andnumerical artifacts in the particle dynamics related to chang-ing the particle representation on the lattice are eliminated. Itis expected that the angular velocity reduces smoothly untilthe particle stops rotating. If the explicit scheme is used, therotational dynamics is very sensitive to the density of theparticle. For low particle density values, there are severe os-cillations in the torque exerted by the fluid and the calcula-tions are not stable. Stability can be recovered only for ex-cessively large particle density values. The improvedtreatments for curved boundaries, while improving the accu-racy of the bounce-back rule, do not reduce the instability inthe rotational dynamics. In contrast, an implicit scheme Ap-pendix allows any particle density to be used and the calcu-lations remain stable. In Fig. 6, we plot the particle angularvelocity as a function of time for a LBM simulation using theimplicit scheme. Because of the inherent complexities in us-ing the implicit scheme for a dense suspension, where par-ticles can interact significantly with each other, we limit mostof the subsequent results to situations where the problemcontains some special symmetry such that rotational dynam-ics of suspended particles can be neglected.

In the second problem inset in Fig. 7, we assign a linearinitial velocity Ut=0=0.05 to a particle located in thefluid such that it is midway between the stationary walls atthe top and bottom of a 101105 lattice. The channel height

eynolds Number ( e )

DragCo

effiient

(CD)

1

10

100

0.1 1 10

Present B (half-ay BB, D 4)Present B (interolated BB, D 4)Feng et al. (14), D 4Present B (half-ay BB, D 30)Suker & Brauer (15), D 30

D

otatio

nalD

ragCo

effiient

(CD)

0.1

1.0

10.0

100.0

0.1 1 10

xat Solution

B

eynolds Number ( e )

FIG. 5. Color online Validation of the LBM model for a thedrag force exerted by a single-phase fluid on a moving cylinder andb the frictional torque on a rotating cylinder.

1 10 100 1000 10000

010

110−

210−

310−

410−

510−

time

Normalized

angularvelocity

Ω(t)

Ω(0)

15

SOLIDLIQUID

FIG. 6. Dynamics of an initially rotating particle of radius R=15 coming to rest within a liquid phase is simulated using theimplicit scheme. The angular velocity is normalized by its initialvalue. LBM parameters used are 300300 periodic lattice, gf =−0.65, gw=−0.04, liq=2.53, and p=5.


066703-9

is 100. The symmetric location of the particle with respect tothe top and bottom walls justifies neglecting the rotationaldynamics and assuming a purely translational motion. It wasconfirmed using the implicit scheme that if the initial angularvelocity is zero, it subsequently remains zero. Ideally, theinitial linear velocity should reduce smoothly to zero becauseof the drag force exerted by the fluid. Figure 7 plots thetransient velocity of the particle normalized using its initialvalue as it moves along the positive x axis for two cases. Thesolid line in Fig. 7 is for a single-phase fluid with a fluiddensity of 2.54. For this case, it is found that the velocity ofthe particle reduces smoothly and exponentially with time.Note that adhesive and cohesive forces are absent for thissingle-phase case. Next, we consider the fluid to be the liquidphase of a multiphase system without the vapor phase andhaving the same density as the single-phase case. This resultis indicated by the data points in Fig. 7. It is seen that al-though the velocity reduces with time with a comparabledecay rate as the single-phase solution, there are consider-able fluctuations. These fluctuations are thought to arise be-cause the cohesive forces between external fluid nodes andadhesive forces between external fluid nodes and particle in-ternal fluid nodes are affected significantly by the changingrepresentation of the particle on the lattice. Reducing oreliminating these fluctuations will be a useful improvementof the present model and is under investigation.

V. RESULTS AND DISCUSSION

A. Single particle at the liquid-vapor interface

We first simulate the dynamics of a single nonrotatingparticle located at a flat liquid-vapor interface Fig. 8. Body

forces like gravity are neglected in this simulation. Becauseof the inherent symmetry, rotational dynamics can be ignoredin the subsequent motion of the particle, if the primary di-rection of motion is perpendicular to the interface. It is wellknown that the interface is a region of minimum energy 41and that a particle suspended on the interface tends to attaina position relative to the interface corresponding to its equi-librium contact angle 42. The equilibrium particle positionrelative to a perfectly flat interface can be uniquely definedby the distance h of the particle center from the interface.Based on particle radius R, the particle contact angle asdefined in Fig. 8 top left can be calculated using =cos−1h /R. This angle can also be estimated by usingYoung’s law Eq. 33. Similar to a liquid drop contacting aflat substrate, the liquid can be said to exhibit wetting behav-ior relative to the particle if /2 and non wetting behav-ior if /2. This wetting property can be controlled by theadhesive force parameter gw, which can be independent ofthat used for fixed walls.

In the LBM simulations, the cohesive and adhesive forceparameters were gf =−0.65 and gw=−0.04, respectively. Theparticle density p=5 and particle radius R=6. The latticesize used was 30100, with a solid wall located at the bot-tom of the domain. This wall was used primarily to anchorthe entire liquid layer on one side. The liquid phase wasinitialized as a 50-unit-thick layer above the solid wall. Theother half of the domain was vapor. The initial location of thecenter of the particle was such that the contact angle was 90°neutral wetting. The relaxation time =1. Periodic condi-tions were used at the left and right boundaries. It is expectedthat the particle will change its vertical position relative tothe interface because of the forces acting on it, eventually,

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000 100000

Single hase

iquid hase

time (t)

010

110−

210−

310−

410−

510−

610−

U(t)

U(t0)

U(t)

FIG. 7. Color online Dynamics of a single particle of radiusR=15 set in motion inside a liquid with an initial linear velocityUt=0=0.05. The solid line represents the velocity decay for thesingle-phase problem and the data points represent the velocity de-cay for the liquid phase with cohesive and adhesive forcespresent. The fluid density in both cases is 2.54 and the particledensity is 5. The particle is initialized at the center of a 100100domain with periodic boundaries on the left and right and solidwalls on the top and bottom.

5

0

5

80

85

0

-0.5

-0.3

-0.1

0.1

0.3

0.5

0 2000 4000 000 8000

Contatangle(θ)

Fy

Fore(F

y)

IQUID

VAPO

SOID

θ h

A B

FIG. 8. Color online Dynamic response of a single particlesuspended at a liquid-vapor interface. The particle is initially placedrelative to the interface such that the contact angle is 90° neutralwetting. The particle then adjusts its vertical position relative to theinterface depending on the adhesive force parameter and oscillatesuntil it attains equilibrium. In this case, the final equilibrium posi-tion of the particle corresponds to 77°. A transient analysis ofthe force acting on the particle reveals two distinct regions: regionA where particle representation on the lattice changes causing largefluctuations in the force and region B, where it no longer changeseliminating force fluctuations.


066703-10

attaining an equilibrium position relative to the interfaceclose to the equilibrium contact angle.

The dynamic response of the particle to the forces actingon it is shown in Fig. 8, which plots the particle contactangle and the force acting on the particle in the vertical di-rection with time. The particle initially oscillates consider-ably in the vertical direction around a position close to theequilibrium position. These oscillations are mainly becauseof the discrete manner in which the particle is represented onthe lattice bottom left in Fig. 8. Eventually, fluid viscositydamps these oscillations and after a certain point in time, theparticle representation on the lattice no longer changes withtime. The LBM calculations then rapidly attain a steadystate. This explains the sudden reduction in force fluctuationsobserved in Fig. 8. For a low particle density, the particleresponds rapidly to the forces acting on it. For a larger par-ticle density, the particle responds more slowly to the forcesexerted on it and the oscillations—while smoother—takelonger to reach a steady value. In this case, the final equilib-rium contact angle of the particle is approximately 77°. Thisresult can be compared with the contact angle for a liquiddrop resting on a flat substrate shown in Fig. 4a. Becausethe same set of adhesive and cohesive force parameters havebeen used in Figs. 4a and 8, the contact angles for thesecases should be the same. To check whether contact anglesbetween these two cases are the same in general, we carriedout LBM simulations for a particle with different wettingstrengths. Figure 9 shows that for the same set of cohesiveand adhesive force parameters, the particle contact angle isidentical to the contact angle for a liquid drop resting on aflat substrate. Thus, the particle contact angle for wettingstrength gw can be deduced from the simpler wetting prob-lem of Fig. 4a. Note that for all particle contact angles, theliquid-vapor interface is perfectly flat when the particlereaches an equilibrium position. This observation is consis-tent with the recent study of Onishi et al. 21, for the case

where the bond number is zero. The effects of gravity havebeen neglected in the present work because the motivationwas to study suspensions in micron-size drops where thebond number is very low.

B. Effect of spurious currents on suspended particlesnear the interface

We next present a case where a single liquid drop and asuspended solid particle are close enough to interact. Theliquid-vapor interface is known to be a region of minimumenergy 41 and thus suspended particles will tend to be at-tracted to it. Because of the symmetry, it is justified to ignorethe rotational dynamics of the particles for the ideal casewhere particles move toward or away from the liquid dropalong a straight line. Spurious velocity currents near the in-terface without suspended particles are first examined fortheir possible role in the particle dynamics close to the inter-face. The LBM simulations are carried out using a 100100 lattice with gw=0 and =1. A circular liquid drop ofradius 25 is centered at 50, 50 and is initially at rest. In Fig.10, the spurious velocity field when the liquid drop reachesequilibrium with its vapor phase is shown for two differentdensity ratios 10 and 30, corresponding to gf =−0.54 and gf=−0.65, respectively, and for two different models, the origi-nal Shan and Chen model 14 and the recent model of Shan28. For all cases, it is found that in the vapor phase, there isa flow toward the liquid drop at the equator and at the northand south poles, while the flow is away from the liquid dropat angles of 45° from the horizontal. The magnitude of thespurious velocity is larger for higher-density ratios. For agiven density ratio, the spurious velocity currents can be re-duced by using the improvements suggested by Shan 28.To demonstrate the effects of spurious currents on suspendedparticles, the case with the maximum spurious currents isselected top right in Fig. 10.

20

40

0

80

100

120

140

20 40 0 80 100 120 140

VAPO

IQUID

1θ

2θ

Contat angle for liquid dro ( θ1 )

Partileontatangle(θ

2)

B result for θ2θ2 θ1

VAPO

IQUID

FIG. 9. Color online Lattice Boltzmann simulation of differentparticle contact angles 2 for a solid particle located at its equi-librium position at a liquid-vapor interface inset, top left. For agiven set of cohesive and adhesive force parameters, the particlecontact angle and the equilibrium contact angle for a liquid drop1 inset, bottom right are almost identical.

Shan

&Ch

en(1

3)Shan

(200

)

Density ratio 10 Density ratio 30

0046.0max

=su

0015.0max

=su 0137.0max

=su

0265.0max

=su

FIG. 10. Color online The spurious velocity field for a liquiddrop solid circle at equilibrium with its vapor phase in the absenceof suspended particles for two different density ratios 10 and 30and using two different multiphase models Shan and Chen 14 andShan 28.


066703-11

In Fig. 11, the liquid drop of radius 25 is initialized at 30,50 in a 101101 lattice with periodic boundaries and al-lowed to attain a steady equilibrium with the vapor phase.Particles are then introduced using two different initial con-figurations. In Fig. 11a, a solid particle particle=5 of radiiR=4.8 is initially located at the coordinates 70, 50 in thevapor phase and is at rest. As the system evolves with time,it is observed the solid particle is attracted toward the liquiddrop. When the liquid drop contacts the moving particle, theparticle is drawn into the liquid-vapor interface and the liq-uid creeps along the particle surface. The initial momentumof the impact leads to some overshoot beyond the equilib-rium contact angle, but the system soon attains a steady statewhere the liquid drop and the solid particle embedded in theliquid-vapor interface both attain a fixed spatial position. Theliquid-vapor interface reattains its circular shape at steadystate. Note that the initial and final momentums of the systemare identical as the liquid drop and particle are at rest at bothtimes. This result seems to be consistent with the validationof the conservation of linear momentum in the LBM algo-rithm. Unfortunately, this seemingly accurate prediction is aresult of the spurious velocity field in the vapor phase, whichis directed toward the liquid drop in the vicinity of the sus-pended particle. If the initial particle location is rotated rela-tive to the center of the liquid drop such that it is at an angleof 45° to the horizontal, the LBM calculations lead to acompletely different result Fig. 11b, where the particle isrepelled from the liquid drop instead of getting attracted to it.Again, this can be understood by looking at the nature of thespurious velocity field outside the curved interface, as shownin Fig. 10.

When the solid particles are initialized such that theytouch the liquid-vapor interface, the particles remain at-tached to the interface. The particle and the liquid drop then

merely adjust their relative positions such that the equilib-rium contact angle is attained. This indicates that the LBM isqualitatively accurate in predicting the interface as a regionof the minimum energy. Removing these unphysical effectsdue to spurious currents is one of the challenges for furtherresearch in this area. In the mean time, using a lower densityratio or implementing the more isotropic model of Shan 28is helpful in reducing the magnitude of these effects.

C. Capillary interactions between particles located on a flatliquid-vapor interface

Next, we examine the behavior of two particles of thesame size that are initially located on a flat liquid-vapor in-terface and are subsequently subjected to different externalforces in the vertical direction. The liquid layer has a depthof 50, with a solid wall close to the bottom of the domain toanchor the liquid layer and the particle diameter is 4.8. Thelattice size used is 100100, with periodic boundary condi-tions on the left and right boundaries. The cohesive forcegf =−0.65 liquid-vapor density ratio 30, the adhesiveforce gw=−0.04, and =1 in both the liquid and vaporphases. The particle density particle=5. The initial conditionspecifies zero velocity for all particles and for the liquid andvapor phases. The two particles are initially located such thattheir equator is slightly below the liquid-vapor interface. Forthe first case shown in Fig. 12a, the initial center to centerspacing between the two particles is 14 and the verticalforces on the two particles 0.06 units each act in oppositedirections, pulling the first particle up toward the vaporphase and the other particle down toward the liquid phase. Itcan be observed that the two particles drift apart under theaction of these opposite forces. Because periodic boundaryconditions are used, the particles spread apart until a roughly

(a)

(b)

FIG. 11. Color online Effect of spurious velocity currents inthe vapor phase on the dynamics of suspended particles filledcircles located close to the liquid-vapor interface of a liquid droplarge circle. a A suspended particle located in the vapor phase atan angular location of 0° relative to the horizontal line from thedrop center is attracted toward the liquid drop and b a suspendedparticle located at 45° is repelled from the liquid drop.

(a) (b) ()

IQUID

VAPO

FIG. 12. Color online LBM simulation of two particles on aninitially flat liquid-vapor interface subject to external forces. aOppositely directed external forces cause the particles to drift apart.b Unidirectional external forces toward vapor phase cause theparticles to move toward each other. c Uni-directionsl forces to-ward the liquid phase cause particles to move toward each other.


066703-12

uniform particle spacing is achieved. In Figs. 12b and12c, the initial center to center spacing between the par-ticles is 40 and the external forces on both particles 0.1 unitact in the same direction. In this case, the particle spacing isobserved to reduce with time and the two particles eventuallyform a closely packed cluster at the center of the domain.

These results can be physically interpreted by imaginingthe liquid-vapor interface as an elastic string that passesthrough both particles while allowing them to slide over thestring. In the first case opposite forces, the external forcespull the particles apart until a minimum-energy configurationis reached where the system is in equilibrium. Similarly, inFigs. 12b and 12c, the minimum-energy equilibrium con-figuration is attained when both particles are almost in con-tact kept slightly apart because of the lubrication andHookean forces and the interface is pulled up and down,respectively, to balance the external forces. The forces actingon the particles have to be smaller than a critical value inorder to avoid detaching the particle from the liquid-vaporinterface. These results are qualitatively consistent with On-ishi et al. 21, who simulated the lateral motion driven bythe capillary interaction between two particles on a binaryinterface. The present model is also able to successfullysimulate the related immersion problem 43, where two par-ticles are attracted toward each other in a thin liquid filmwetting a substrate. The effect of rotational dynamics onthese results needs to be included for a more realistic simu-lation.

D. Spinodal decomposition in the presence of suspendedparticles

The effect of suspended particles on the spinodal decom-position process was recently studied by Stratford et al.19,20 in the context of demixing of a binary liquid mixture.Interesting effects, such as colloidal jamming, have been ob-served, where the suspended particles are sequestered in thefluid-fluid interfacial region and thus prevent or markedlyreduce the decomposition of a binary mixture. We have car-ried out a similar study for the present multiphase systemusing a 300300 lattice with periodic boundaries. It is nec-essary to use virtual images of each particle in adjacent pe-riodic domains during the calculations. This is required tocheck if a pair of particles is close enough to interact acrossperiodic boundaries or when a particle intersects with theedges of the lattice and consequently has to reappear, in part,at the opposite edge. The initial condition is an array of 36suspended particles of radius R=4.8 3% volume fractionarranged in the form of a regular equally spaced 66 arrayof rows and columns Fig. 13. All particles are initially atrest and are surrounded by a fluid phase that is initially at auniform density of 0.693. Other parameters used are gf =−0.65 liquid-vapor density ratio 30, gw=−0.04 corre-sponding to a 77° equilibrium contact angle, =1, andparticle=5. At time t=0, the fluid density is randomly per-turbed by 0.5% and these perturbations then grow becauseof the mutual cohesive attraction and the fluid begins toseparate into distinct liquid and vapor phases. However, thecoarsening process driven by the tendency of reducing the

liquid-vapor interfacial area is inhibited because of the sus-pended particles compared to the coarsening process withoutparticles not shown. These particles move to the liquid-vapor interfacial regions and several liquid drops with par-ticles along the interface region can be observed. In the ab-sence of Brownian motion, once trapped, particles willremain at the liquid-vapor interface during the coarseningprocess. Eventually, some of these liquid drops coalesce to-gether, causing a further rearrangement of the particles onthe interface. The accumulation of particles at the interfaceand the curtailment of coarsening observed here are verysimilar to those described in Stratford et al. 20 in the con-text of a binary mixture. However, unlike the neutrally wet-ting particles used in Ref. 20, the particles in the presentstudy preferentially wet the liquid phase. Thus, qualitativelyaccurate results are obtained even when the rotational dy-namics of particles is neglected. The effect of rotational dy-namics should, however, be included for a more accuratesimulation.

VI. CONCLUDING REMARKS

In this paper, a two-dimensional multiphase LBM hasbeen developed such that one or both phases liquid andvapor can have suspended solid particles dispersed within it.The suspended particles are circular in shape and are dy-namically coupled to the fluid flow. The Shan and Chenmodel 14 has been used to simulate the two-phase flow andthe particle suspension model pioneered by Ladd and co-workers has been modified to account for adhesive forcesbetween the suspended particles and the liquid and vaporphases. The combined model has been extensively validatedand found to be accurate and physically realistic, althoughthere are some instability issues for two-phase scenarios re-lated to force fluctuations when suspended particles move onthe LBM lattice. Recently, an improved method for initializ-

t 100 t 1000 t 5000

t 20000 t 50000 t 85000

FIG. 13. Color online LBM simulation of spinodal decompo-sition in the presence of suspended particles. The particles are ini-tially arranged in a uniform 66 array and are at rest. Subse-quently, as the green light gray liquid domains begin to form andcoalesce, the particles tend to accumulate at the liquid-vapor inter-face and inhibit coarsening of the interface.


066703-13

ing new fluid nodes uncovered by moving particles and acorrected momentum exchange algorithm for calculatingforces on suspended particles have been proposed 44,45.These schemes do not consider adhesive and cohesive forces,but they might be useful in reducing the instability and arepresently under investigation.

The results demonstrate the capability of the multiphaseLBM to model complex fluid dynamic problems involvingsuspended solids. It is shown how a suspended particle lo-cated at the liquid-vapor interface attains a stable positioncorresponding to its wetting properties. The LBM confirmsthat the interface is a local energy minimum for suspendedparticles. Spurious currents have been shown to lead to un-physical effects, such as attraction or repulsion of freely sus-pended particles in the vapor phase to a liquid drop, depend-ing on the angular position of the particle relative to thedrop. These effects can be reduced, but not completely elimi-nated, by reducing the density ratio or by using more isotro-pic models 28. Particle accumulation at the interface ob-served in the context of multicomponent fluid mixtures isalso observed in the present multiphase scenario, as illus-trated by the case of the spinodal decomposition in the pres-ence of suspended particles.

The entire algorithm can be efficiently parallelized andextended to three dimensions. In many applications of col-loidal drops settling onto a substrate, such as ink-jet-printedelectronics, the suspended particle dynamics is influenced byflow fields within the liquid due to thermal gradients andbecause of additional long-range forces between the particlesthemselves. The effect of particle rotation dynamics alsoneeds to be analyzed more carefully. Adding in these effectsis a part of our ongoing effort and the results will be reportedin a future publication.

ACKNOWLEDGMENTS

The authors would like to thank the National ScienceFoundation Grant No. CAREER-0846825, Center for Ad-vanced Microelectronics Manufacturing CAMM at theState University of New York at Binghamton, and AmericanChemical Society Petroleum Research Fund Grant No.47731-G9 for providing financial support.

APPENDIX: IMPLICIT SCHEME FOR UPDATINGLINEAR AND ANGULAR VELOCITY OF A SINGLE

SUSPENDED PARTICLE

In this appendix, we extend and adapt the implicit schemeof Lowe et al. 6 to the particle suspension model describedin this work. The basic idea is that the unknown particlevelocity at the new time level is to be used in the expressionfor calculating the boundary velocity of the particle given byEq. 24. Expressions for the forces and torques acting on theparticle of mass M and moment of inertia I are first derivedusing the two unknown velocity components along x and ydirections at the new time level denoted by u and v, respec-tively, and the unknown angular velocity of the particlealong z direction at the new time level denoted by . New-ton’s second law is then applied to relate these forces along

x and y and torque along z to the change in the linear andangular velocities of the particle. The result is a set of threelinear equations that can be solved for the three unknowns.After some algebra, the resulting system of equations can bewritten as

A11 A12 A13

A21 A22 A23

A31 A32 A33 u

v

= B1

B2

B3 . A1

The elements of the coefficient matrix are given by

A11 = 1 +6

Mbn

wexex, A2

A12 = A21 =6

Mbn

weyex, A3

A13 = −6

Mbn

rywexex − rxweyex , A4

A22 = 1 +6

Mbn

weyey , A5

A23 = −6

Mbn

rywexey − rxweyey , A6

A31 = −6

Ibn

rywexex − rxweyex , A7

A32 = −6

Ibn

rywexey − rxweyey , A8

A33 = 1 +6

Ibn

rxrxweyey + ryrywexex

− 2rxrywexey . A9

The right-hand side of Eq. A1 is calculated using

B1 = u0 +2

Mbn

f+ex +

1

Mbn

Fadh,xp +

1

MFU,x + FC,x ,

A10

B2 = v0 +2

Mbn

f+ey +

1

Mbn

Fadh,yp +

1

MFU,y + FC,y ,

A11


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B3 = 0 +2

Ibn

rxf+ey − ryf

+ex

+1

Ibn

rxFadh,y − ryFadh,x

−1

IU

rU,xUuU,y − rU,yUuU,x

+1

IC

rC,xCuC,y − rC,yCuC,x . A12

In the above expressions, u0, v0, and 0 represent knownvalues at the current time level. The summations are to becarried out over each boundary node bn and over all therelevant directions at that boundary node. The location ofthe relevant node point is denoted by r. Subscripts x and ydenote components along the x and y axes and subscripts Uand C denote uncovered and covered nodes, respectively.Implementing this scheme requires two passes through thedomain. During the first pass, the coefficients Aij and Bi are

assembled. After the linear system of equations, i.e., Eq.A1 is solved, another pass through the domain is made toupdate the relevant PDFs based on the calculated values of u,v, and . The adhesive forces in Bi are based on Eq. 28 andthe forces because of changing particle representation on thelattice are calculated using Eqs. 17–20. The advantage ofusing the implicit scheme is that rotational dynamics can beincluded in the calculations without introducing the numeri-cal instability. The scheme described here can be imple-mented for any number of suspended particles as long as theparticles do not come too close to one another. If lubricationand Hookean forces between particles are to be included inthe derivation of the implicit scheme, the implementation ismuch more complex due to the increase in the number ofunknowns induced by particles that are close enough to in-fluence each other see Nguyen and Ladd 5 for more de-tails. Another useful extension will be to incorporate inter-polated bounce back into the implicit scheme for accuratelydescribing interactions with curved boundaries. These exten-sions are still in progress and will be reported in a futurepublication.

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multiphase lattice boltzmann method for particle...

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