Displacement of a two-dimensional immiscible droplet in … · Displacement of a two-dimensional...

PHYSICS OF FLUIDS VOLUME 14, NUMBER 9 SEPTEMBER 2002

Displacement of a two-dimensional immiscible droplet in a channelQinjun KangLos Alamos National Laboratory, Los Alamos, New Mexico 87545and The Johns Hopkins University, Baltimore, Maryland 21218

Dongxiao ZhangLos Alamos National Laboratory, Los Alamos, New Mexico 87545

Shiyi ChenThe Johns Hopkins University, Baltimore, Maryland 21218and Peking University, Beijing, China

~Received 4 December 2001; accepted 17 June 2002; published 5 August 2002!

We used the lattice Boltzmann method to study the displacement of a two-dimensional immiscibledroplet subject to gravitational forces in a channel. The dynamic behavior of the droplet is shown,and the effects of the contact angle, Bond number~the ratio of gravitational to surface forces!,droplet size, and density and viscosity ratios of the droplet to the displacing fluid are investigated.For the case of a contact angle less than or equal to 90°, at a very small Bond number, the wet lengthbetween the droplet and the wall decreases with time until a steady shape is reached. When the Bondnumber is large enough, the droplet first spreads and then shrinks along the wall before it reachessteady state. Whether the steady-state value of the wet length is greater or less than the static valuedepends on the Bond number. When the Bond number exceeds a critical value, a small portion ofthe droplet pinches off from the rest of the droplet for a contact angle less than 90°; a larger portionof the droplet is entrained into the bulk for a contact angle equal to 90°. For the nonwetting case,however, for any Bond number less than a critical value, the droplet shrinks along the wall from itsstatic state until reaching the steady state. For any Bond number above the critical value, the dropletcompletely detaches from the wall. Either increasing the contact angle or viscosity ratio ordecreasing the density ratio decreases the critical Bond number. Increasing the droplet size increasesthe critical Bond number while it decreases the critical capillary number. ©2002 AmericanInstitute of Physics.@DOI: 10.1063/1.1499125#

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I. INTRODUCTION

Multiphase or multicomponent flow in porous mediaan important phenomenon for a wide range of practical prlems, including oil recovery and the transportnonaqueous-phase liquid contamination in the soil via waor surfactant flooding. The study of the displacement of imiscible fluids in a single pore throat is a prerequisitesimulating flow in porous materials, which consist of neworks of pore throats and pore bodies. Previous workimmiscible displacement in capillary tubes has focusedthe displacement of a fluid meniscus,1 the displacement ofreely suspended fluid droplets,2–4 and the displacement ofdroplet attached to the wall where the no-slip boundary cdition is enforced and the contact lines remain pinned to thinitial positions.5–9

Most immiscible displacement-related problems in caillary tubes are a moving contact-line problem, which is important both from a purely theoretical and a practical poinview, and has attracted the attention of mainvestigators.1,10–15A direct application of classical hydrodynamics on the liquid/liquid/solid systems encounters funmental difficulties. Assumptions need to be made to descthe behavior of the dynamic contact angle and to removenonintegrable stress singularity at the dynamic conlines.15

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To overcome these problems, Shikhmurzaev developgeneral mathematical model that describes, in the cassmall capillary number and small Reynolds number, the mtion of an interface between immiscible viscous fluids aloa smooth homogeneous solid surface.15 Molecular dynamicssimulations are also used to study the immiscible flusystem.14 Koplik’s simulations confirm Dussan’s theoreticpredictions of the breakdown of the no-slip condition.10 Thelattice Boltzmann ~LB! method, a promising numericamethod, is also applied to the moving contact-liproblem.16–18Although different methods have been applito study a variety of immiscible fluid systems, the displacment of an immiscible droplet adhering to a wall in a chanunder external forces has received less attention. This plem is particularly interesting and challenging because this a coalescence of contact lines when the droplet detafrom the wall.

Using a boundary-integral method, Schleizer aBonnecaze19 studied the dynamic behavior and stability oftwo-dimensional immiscible droplet subject to shearpressure-driven flow under conditions where inertial agravitational forces can be neglected. The droplet is attacto the solid surface, and the two contact lines are either fior mobile. To allow the droplet to slip along the wall, theapplied an integral form of the Navier-slip model whi

3 © 2002 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp

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3204 Phys. Fluids, Vol. 14, No. 9, September 2002 Kang, Zhang, and Chen

keeping the contact angle fixed. They investigated the effof contact angle, capillary number~the ratio of viscous tosurface forces!, droplet size, and viscosity ratio on the drolet behavior. Their simulation results showed that for contangles less than or equal to 90°, a stable droplet sprealong the wall until a steady shape is reached; the drothen moves across the wall at a constant velocity.

For contact angles greater than 90°, the wet lengthtween a stable droplet and the wall decreases until a steshape is reached. They also claimed that above a cricapillary number, the droplet completely detaches for a ctact angle of 120°. However, they indicated that it wasdifficult to include the coalescence of the contact linestheir numerical method. In this respect, the LB method mbe an alternative numerical approach that is capable of nrally simulating the coalescence of contact lines.20,21

Unlike conventional numerical schemes based oncretizations of macroscopic continuum equations, themethod is based on microscopic models and mesoscopinetic equations,22 giving the LB method the advantage in thstudy of nonequilibrium dynamics, especially in fluid-floapplications involving interfacial dynamics and complboundaries~geometries!. The LB method has also proved tbe competitive in studying a variety of flow and transpphenomena~see Refs. 22–26 for reviews!, including flow orchemical dissolution in porous media.27,28One particular ap-plication of the LB method that has attracted consideraattention is modeling the flows of multiphase or multicomponent fluids. These flows are important, but because ofcomplex interfaces in the inhomogeneous flows, they areficult to simulate with the conventional technique of solviNavier–Stokes-type macroscopic equations. The LB methhowever, because of its kinetic nature, can handle these plems more easily.22 Although the LB method cannot simulatthe moving contact-line problem on the microscopic scalemolecular dynamics does, it can simulate physics on a ssmaller than the macroscopic methods based on the NavStokes-type equations and can simulate mesoscopic-physics, which may be important in this problem. In simlating the immiscible displacement of a droplet with the Lmethod, no assumption about the relationship betweencontact angle and velocity of the contact line is necessand the coalescence of the contact lines can be demonstnaturally.

There are four LB models used for the study of muphase and/or multicomponent flow. The first model, usinrecoloring process,29,30 is quite robust and its algorithm istraightforward. However, it has two drawbacks: first, trecoloring process requires time-consuming calculationsthe local maxima; second, the perturbation step with thedistribution of colored distribution functions causes an anitropic surface tension that induces unphysical vortices ninterfaces.22 The second model makes use of an interpartpotential.31–34 This model has a correct physical basis agives a nonideal equation of state. The isotropy of the sface tension at the interface is improved.35 The separation ofa two-phase fluid into its components is automatic.22 Thethird model, with free energy,36,37 is the first LB two-phasethermal model, and it also has a sound physical basis


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produces a correct equation of state and has a consisurface tension representation. However, it does not saGalilean invariance. Its macroscopic equation has a velocdependent pressure.38 The fourth model can be consideredbe a significant modification of the second model.39 In addi-tion to the attractive force, a repulsive force is included. Tnumerical stability is improved by reducing the effect of nmerical errors in the calculation of molecular interactionAn index function is used to track the interfaces betwedifferent phases. All of the above models have been succfully used in various applications, including flow througporous media40–43 and Rayleigh–Taylor instability.39,44

In this paper, we use the model with interparticle potetial. Besides the above-mentioned reasons, this model isvenient for implementing different wettabilities and handlinfluids with different densities and viscosities. With thmodel, we have studied the dynamic behavior of a twdimensional immiscible droplet in a simple channel subjto gravitational forces and assessed the effects of the conangle, Bond number, droplet size, and the density andcosity ratios of the droplet to the displacing fluid.

II. MODEL AND THEORY

A. Multiphase or multicomponent lattice Boltzmannmodel

Let us first review the multiphase or multicomponent Lmodel proposed by Shan and Chen.31,32 Their model allowsfor density distribution functions of an arbitrary numbercomponents with different molecular mass. The interactbetween the particles is included in the kinetics through aof potentials. The LB equations for thekth component can bewritten in the following form:

f ik~x1eid t ,t1d t!2 f i

k~x,t !52f i

k~x,t !2 f ik~eq!~x,t !

tk, ~1!

where f ik(x,t) is the number density distribution in thei th

velocity direction for thekth fluid at positionx and timet,andd t is the time increment. On the right-hand side,tk is therelaxation time of thekth component and is in the latticunit, and f i

k(eq)(x,t) is the corresponding equilibrium distribution function. For a two-dimensional 9-speed LB mod~D2Q9, where D is the dimension and Q is the numbervelocity directions!, f i

k(eq)(x,t) has the following form:45,46

f 0f ~eq!5aknk2

2

3nkuk

eq•uk

eq,

f ik~eq!5

~12ak!nk

51

1

3nk~ei•uk

eq!

11

2nk~ei•uk

eq!221

6nkuk

eq•uk

eq for i 51,...,4, ~2!


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3205Phys. Fluids, Vol. 14, No. 9, September 2002 Displacement of a 2D immiscible droplet

f ik~eq!5

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eq for i 55,...,8.

In the above equations,ei ’s are the discrete velocities, whicare chosen to be

ei550, i 50,

S cos~ i 21!p

2,sin

~ i 21!p

2 D , i 5124,

&ScosF ~ i 25!p

21

p

4 G ,sinF ~ i 25!p

21

p

4 G D , i 5528,

~3!

ak is a free parameter, which relates to the sound speedregion of purekth component as (cs

k)25 35(12ak);

46 nk

5( i f ik is the number density of thekth component. The

mass density of thekth componentrk is defined asrk

5mknk5mk( i f ik , and the fluid velocity of thekth fluid uk is

defined throughrkuk5mk( iei f ik , wheremk is the molecular

mass of thekth component. The parameterukeq is determined

by the relation

rkukeq5rku81tkFk , ~4!

where u8 is a common velocity on top of which an extrcomponent-specific velocity due to interparticle interactionadded for each component, andFk5F1k1F2k1F3k is thetotal force acting on thekth component, including fluid–fluidinteraction F1k , fluid–solid interactionF2k , and externalforce F3k .42 To conserve momentum at each collision in tabsence of interaction~i.e., in the case ofFk50!, u8 has tosatisfy the relation

u85S (k51

srkuk

tkD Y S (

k51

srk

tkD . ~5!

The interactive force between particles of thekth componentat sitex and thek̄th component at sitex8 is assumed to beproportional to the product of their ‘‘effective number desity’’ ck(nk) defined as a function of local number densiThe total fluid–fluid interactive force on thekth componentat sitex is

F1k~x!52ck~x!(x8

(k̄51

s

Gkk̄~x,x8!c k̄~x8!~x82x!, ~6!

whereGkk̄(x,x8) satisfiesGkk̄(x,x8)5Gkk̄(x8,x), andck(x)is a function ofx through its dependency ofnk . The aboveform of interaction potential was originally used for singlspeed lattices such as the two-dimensional hexagonal anfour-dimensional FCHC,31 and only homogeneous isotropinteractions between the nearest neighbors were considThe interaction potential of the D2Q9 lattice can be obtainwith a similar method adopted by Martys and Chen42 toproject the interaction potential from 4D FCHC lattice to tD3Q19 model. In this case, the nearest-neighbor interac


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in four dimensions corresponds to a potential that coupnearest and next-nearest neighbors in the D2Q9 latmodel. In this case,

Gkk̄~x,x8!5H gkk̄ , ux2x8u51,

gkk̄/4, ux2x8u5&,

0, otherwise.

~7!

Heregkk̄ is the strength of the interparticle potential betwecomponentk and k̄. The effective number densityck(nk) istaken asnk in this study. Other choices will give a differenequation of state.

At the fluid/solid interface, the wall is regarded asphase with a constant number density. The interactive fobetween the fluid and wall is described as

F2k~x!52nk~x!(x8

gkwnw~x8!~x82x!, ~8!

wherenw is the number density of the wall, which is a costant at the wall and zero elsewhere, andgkw is the interac-tive strength between componentk and the wall.gkw is posi-tive for a nonwetting fluid and negative for a wetting fluiBy adjusting it, we can get different wettabilities.

The action of a constant body force can be simply intduced as

F3k5rkg5mknkg, ~9!

whereg is the body force per unit mass.The Chapman–Enskog expansion procedure can be

ried out to obtain the following continuity and momentuequations for the fluid mixture as a single fluid:33

]r

]t1¹•~ru!50, ~10!

rF]u

]t1~u•¹!uG52¹p1¹•@rn~¹u1u¹!#1rg, ~11!

wherer5(krk is the total density of the fluid mixture, anthe whole fluid velocity u is defined by ru5(krkuk

1 12(kFk .34 Notice that the introduction of fluid–solid inter

action has no effect on the macroscopic equations sinceF2k

only exists at the solid/fluid interface.Since the Shan–Chen model was proposed, it has b

widely used by various investigators. Houet al.35 performeda static bubble test with this model. Their results show thaphysically correct isotropic, stable bubble satisfyinLaplace’s law can be simulated. Fanet al.16 applied themodel to the simulation of contact line dynamics in a twdimensional capillary tube. In their paper, the velocity dpendency of the contact angle is studied for two differewetting cases. Their simulation results are in good agreemwith those based on theoretical computations and with mlecular dynamics simulations. Martys and Chen42 used thismodel to simulate the multicomponent fluid flow in complethree-dimensional geometries. In their study, the displament of one fluid by another in a porous medium was meled. The relative permeability for different wetting fluisaturations of a microtomography-generated image of sastone was calculated and compared favorably with exp


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ment. Sehgalet al.47 developed a FlowLab code basedthis model and performed a series of multiphase flow benmark tests, including a static bubble test, a phase transtest, and the test of agglomeration of two immiscible fluiThey then applied this method to the simulation of dropdeformation and breakup. The deformation/breakup regmap, derived from their simulation results, is in quite goagreement with experimental data. All the numerical simutions show that the hydrodynamics from this method are crect.

B. Application to systems of immiscible fluids withdifferent densities and viscosities

The previous applications of the above LB model to stems of immiscible fluids considered only fluids with thsame molecular mass and viscosity.16,35,42,47Our study, how-ever, aims to study the effects of density and viscosity rabetween the displaced and displacing fluids.

When the above equilibrium distribution function, E~2!, is chosen, in a region of purekth component, the pressure is given bypk5(cs

k)2rk5(csk)2mknk , where (cs

k)2

5 35(12ak). To simulate a multiple component fluid wit

different component densities, we let (csk)2mk5c0

2, wherec02

is a constant. In this study,c025 1

3, andmk>1. Then the pres-sure of the whole fluid is given by p5c0

2(knk

1 32(k,k̄gk,k̄ckc k̄ , which represents a nonideal gas law. T

viscosity is given byn5 13((kbktk2 1

2), where bk is themass density concentration of thekth component and is defined asrk /(krk .35

III. SIMULATION RESULTS AND DISCUSSION

The two-dimensional geometry used in our simulatiois shown in Fig. 1, where an immiscible droplet~fluid 2! withareaA is put into a two-dimensional channel filled with flui1. The lettersl andh are the length and width of the channerespectively;b0 is the wet length between the droplet and twall; a0 is the droplet height; andu1 andu2 are the contactangles of fluid 1 and 2, respectively. The contact angledefined as the angle between a two-fluid interface and a ssurface. As shown in Fig. 1, each fluid has its own contangle and the sum of the two must equal 180°. The wetfluid ~the fluid that tends to wet the surface! has a contactangle of less than 90°, and the nonwetting fluid~the fluid thathas less affinity for the solid surface! has a contact angle ogreater than 90°.

From simple geometry,R, the radius of the droplet, cabe calculated48 from the final steady values ofa0 andb0 as

R5a0

21

b02

8a0. ~12!

The contact angle of the droplet satisfies the followirelation:48

tan~u2!5b0

2~R2a0!. ~13!

The area, the radius, and the contact angle of the drosatisfy the following equation:


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let

A5R2~u220.5 sin~2u2!!. ~14!

For simplicity, hereafter we refer tou2 when we saycontact angle oru. The coordinate origin is located at thbottom left corner. A particle distribution function bouncback scheme49,50 is used at the walls to obtain no-slip veloity conditions. By the so-called bounce-back scheme,mean that when a particle distribution streams to a wall nothe particle distribution scatters back to the node it ca

FIG. 1. Schematic illustration of simulation geometry.

FIG. 2. The dependency of the contact angle of the droplet,u, on theinterparticle potential between the droplet and the wall,g2w . Other param-eters arer2 /r151, n2 /n151, g125g2150.2, andA/h250.31.


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from. Periodic boundary conditions are applied aty50 andy5 l . The flow is induced by a constant body force along2y direction.

A. Evaluation of contact angle

The static contact angle can be reasonably well predicby Eq.~8!.42 In this simulation, no body force is applied, anl 5300, h540, both in lattice unit spacings,A/h250.31,r2 /r151, n2 /n151, g1w52g2w , g115g2250, and g12

5g2150.2. Initially, b052a052R. When a static droplet isobtained, the values ofa0 andb0 are measured, andR anduare calculated from Eqs.~12! and~13!, respectively. The val-ues ofg1w andg2w are varied to obtain steady droplets widifferent contact angles. The contact angle is a linear fution of g2w , which is in agreement with the work of Yanet al.51 As shown in Fig. 2,u is less than 90° wheng2w isnegative, which means fluid 2 tends to wet the surface;u isequal to 90° wheng2w equals zero, indicating that neither othe fluids has the preference to wet the surface; andu isgreater than 90° wheng2w is positive, meaning that the fluid2 is nonwetting to the wall. Hence by adjustingg2w , we mayobtain different contact angles, i.e., different wettabilities.shown in Fig. 3, contact angles 78°, 90°, and 118° aretained by lettingg2w equal20.02, 0, and 0.05, respectivel

B. Evaluation of surface tension

Previous tests of Laplace’s law with this LB model weperformed in situations where periodic boundary conditiowere applied in all directions.31,35,42,47In this study, however,we performed the bubble test with the existence of the wto test the correctness of the method handling the fluid–flas well as the fluid–solid interactions. Laplace’s law48 states

pi2po5s

R, ~15!

FIG. 3. Three static contact angles obtained by adjustingg2w , the param-eters are the same as in Fig. 2:~a! g2w520.02, u578°, ~b! g2w50, u590°, ~c! g2w50.05,u5118°.


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wherepi andpo are the pressure inside and outside the drlet, respectively;s is the surface tension between fluid 1 a2; andR is the radius of the droplet.

In this simulation, there is no body force applied, andl5300, h5150, fluid 1 and 2 have the same density aviscosity,g115g2250, andg125g2150.2. The values ofg1w

andg2w , as well as the initial values ofa0 andb0 , are variedto obtain steady droplets with different radii and contaangles. The final droplet radius and pressure differencetween the inside and outside of the droplet are measuSteady-state pressure inside the droplet (pi) and outside thedroplet (po) is measured on nodes that are away frominterface since the values of the pressure vary nearinterface.35 These results are listed in Table I and plottedFig. 4. The droplet radius, pressure difference, and calculasurface tension are all in lattice units and can be relatedpractical problems. The square symbols in Fig. 4 indicateresults of the LB simulations, and the solid line is the leasquare linear fit. The slope~i.e., the surface tension! is0.078 94; the intercept is 0.000 0176; and the coefficiendetermination is 0.999 85. It is clear that all points fitstraight line through the origin quite well. The pressure dference inside and outside the bubble is indeed proportioto the reciprocal of the radius. It is clear that Laplace’s lawwell satisfied in the bubble test simulations with the extence of the wall. The agreement between the theory andnumerical simulation indicates that the described LB meth

FIG. 4. Test of Laplace’s law, the slope~i.e., the surface tension! is0.078 94; the intercept is 0.000 0176; and the coefficient of determinatio0.999 85.

344355152

342163016

TABLE I. Test of Laplace’s law.

Radius 8.145 079 10.673 730 13.060 358 15.165 089 18.432 281 23.592Dp 0.009 636 0.007 469 0.006 107 0.005 249 0.004 308 0.003s 0.078 486 0.079 722 0.079 760 0.079 601 0.079 406 0.079Radius 28.679 053 33.801 003 38.815 093 48.828 534 58.803 024 68.801Dp 0.002 757 0.002 341 0.002 040 0.001 626 0.001 355 0.001s 0.079 068 0.079 128 0.079 183 0.079 395 0.079 678 0.080



FIG. 5. Evolution of the dimensionless wet lengthb/b0 between the wall and the sliding droplet at three contact angles~u578°, 90°, and 118°!, whereb0

is the wet length at time 0. Time is made dimensionless by characteristic timel /U. Other parameters are the same as in Fig. 2:~a! B050.061 42,~b! B0

50.3685,~c! B050.6044,~d! B050.8599.

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handles the fluid–fluid interaction and fluid–solid interacticorrectly.

C. Displacement of droplet

In the following simulations,l is 300 andh is 40. After astatic droplet is achieved at time zero, a constant body fo~gravitational force! is applied to both the displacing fluiand the droplet along the2y direction. The simulation goeon until steady state is reached or part of the droplet pincoff from the rest, or the entire droplet detaches from the wThe steady state here is defined as the state at whichdroplet moves along the wall with a constant velocity andshape does not change with time. Numerically, the stestate is reached when the wet length between the dropletwall varies less than 0.1% in 1000 successive time steps.effects of the static contact angle, the Bond number,droplet size, and the density and viscosity ratios betweentwo fluids are investigated. In what follows, we defineU


e

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he

5r1gh2/m1 as the characteristic velocity, andl /U as the char-acteristic time, wherer1 andm1 are the density and viscositof fluid 1, respectively. The Bond number is defined asB0

5r2gA/s, wherer2 is the density of fluid 2, andg is thegravitational factor. The Bond number is a dimensionleparameter that indicates the ratio of gravitational forcesurface force.

1. Effect of contact angle

Figure 5 shows the time evolution of the dimensionlewet lengthb/b0 between the wall and the sliding droplewith r2 /r151, n2 /n151, andA/h250.31, at three contacangles ~u578°, 90°, and 118°! and four Bond numbers~0.061 42, 0.3685, 0.6044, and 0.8599!. Different Bond num-bers are obtained by changingg, the gravitational factor. At avery low Bond number of 0.061 42@Fig. 5~a!#, b/b0 de-creases with time for all contact angles before it reachesteady-state value, and the three curves are very close to


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other. At a moderate Bond number of 0.3685@Fig. 5~b!#, thethree curves begin to separate. For bothu578° ~wettingcase! and u590°, b/b0 increases with time at first, thedecreases. The steady-state value ofb/b0 is greater than 1 inthe former case and smaller than 1 in the latter case. Whethe steady-state value ofb/b0 is greater or less than 1 depends on the Bond number~see Fig. 10 for details!. For thecases ofu5118° ~nonwetting case!, b/b0 dramatically de-creases with time and reaches 0 at time 6, when the drototally detaches from the wall. In this nonwetting case,Bond number already exceeds a critical value called critBond number (B0

c), below which a sliding droplet with aconstant shape may be observed, and above whichsteadily sliding droplet will be obtained. At a high Bonnumber of 0.6044, as shown in Fig. 5~c!, in the wetting caseb/b0 increases rapidly at first and then gradually decreabefore it plateaus to a value of about 1.26. The nonwetdroplet again dramatically shrinks along the wall with timand detaches from the wall at time 6.

The behavior of the droplet withu590°, however, ismore complicated. It spreads along the wall at first, thshrinks. Its upper portion deforms and moves faster thanlower portion and eventually pinches off. Hence this Bonumber also exceeds the critical value of the droplet incase. The highest Bond number of 0.8599 is greater thancritical Bond number of all three cases; thus no steadily sing droplet will be observed. However, the droplet with dferent static contact angles behaves very differently. Wu578°, the droplet spreads along the wall continuouslyfore a portion of it pinches off and is entrained into the buflow. When u590°, the droplet spreads along the wallfirst and then shrinks, and the upper portion of the droppinches off when the droplet is shrinking. Whenu5118°,the downstream contact line moves more slowly thanupstream contact line all the way, and the contact lines evtually come together; thus the entire droplet detaches fthe wall.

The dynamic process of the droplet with the three ctact angles detaching from the wall at Bond number 0.85is shown in Fig. 6. It is clear from Fig. 6 that the detachportion of the droplet atu590° is larger than that atu578°, which confirms the assertion by Schleizer aBonnecaze19 that above the critical capillary number, increasing the contact angle results in a larger fraction of drlet being entrained in the bulk. Figure 7 shows the corsponding velocity fields of the detaching process. We canthat whenu578° and 90°, the velocity inside the upper potion of the droplet is larger than that inside the lower portiowhich causes the upper portion to translate more rapidlyeventually pinches off from the lower portion. Whenu5118°, however, the velocity inside the droplet is almouniform and larger than that of fluid 1 near the wall. Thmeans the shear forces between the droplet and the wavery large and eventually cause the entire droplet to defrom the wall. Figure 8 shows the velocity fields when tdroplet is steadily sliding along the wall. Note that the nslip boundary condition is well satisfied. The velocity fierelative to the average droplet velocity reveals, in Figs. 8~d!–8~f!, a vortex flow pattern within the droplet itself, which


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-very similar to that obtained by Grubert and Yeomans.18

From the above discussion, we notice that the criticBond number ofu590° is smaller than that ofu578° butgreater than that ofu5118°. Figure 9 shows a general relationship between the critical Bond number and the static c

FIG. 6. Dynamic behavior of the droplet atB050.8599, corresponding toFig. 5~d!: ~a! u578°, a small portion of the droplet pinches off from threst. At t522.4, the detached portion re-enters the system due to the podic conditions applied alongy direction,~b! u590°, a large portion of thedroplet pinches off from the rest,~c! u5118°, the entire droplet detachefrom the wall.


g


FIG. 7. Velocity fields near the droplet, correspondinto Fig. 6.

s

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s- se.

tact angle. It is clear that the critical Bond number decreaas the contact angle increases.

2. Effect of Bond number

Figure 10 illustrates the dependency of a steady-svalue of the nondimensional wet lengthb/b0 on the BondnumberB0 , with a density ratior2 /r151, viscosity ration2 /n151, droplet sizeA/h250.31, and three contact angle~u578°, 90°, and 118°!. Different Bond numbers are ob


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tained by changingg, the gravitational factor. At a low Bondnumber,b/b0 decreases with the increase ofB0 in all cases.At B050.1843, for the wetting case,b/b0 reaches its mini-mum and begins to increase, whileb/b0 in the other caseskeeps decreasing. AsB0 exceeds 0.2439~the critical Bondnumber for the nonwetting droplet withu5118°!, the non-wetting droplet totally detaches from the wall~see Figs. 9and 6 for details!. When B0 is about 0.2457,b/b0 for thecase ofu590° reaches its minimum and begins to increa


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Figure 11 shows the dependency of the steady-svalue of the dimensionless contact-line velocitymVcl /s ~orthe droplet velocity, since the shape of the droplet doeschange with time any more! on the Bond number. Foru590° oru5118°, this velocity is linear inB0 , meaning that

FIG. 8. Velocity fields near the steadily sliding droplet:~a! velocity profilenear the droplet with static contactu578° atB050.7985,~b! velocity pro-file near the droplet with static contactu590° atB050.5835,~c! velocityprofile near the droplet with static contactu5118° atB050.2150,~d! ve-locity profile within the droplet for case~a! minus average droplet velocity~e! velocity profile within the droplet for case~b! minus average droplevelocity, ~f! velocity profile within the droplet for case~c! minus averagedroplet velocity.

FIG. 9. Dependency of the critical Bond number (B0c) on the contact angle

~u!. Other parameters are the same as in Fig. 2.


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the droplet translates more rapidly asB0 increases. The slopefor u590° is smaller than that foru5118°. For the wettingcase, however, there are two regions where the contactvelocity varies linearly. The transition between these regiois 0.4300<B0<0.6142, which corresponds to the regiowhere b/b0 increases rapidly, as seen in Fig. 10, and thcorresponds to the region where the droplet shape chasignificantly, as seen in Fig. 12~a!. Figure 12 shows thesteady-state shape of the droplet at different Bond numb

3. Effect of droplet size

Figure 13 shows the dependency of the critical Bonumber on the dimensionless areaA/h2 of the droplet, atu578°, 90°, and 118°. The density and viscosity ratios b

FIG. 10. Dependency of the steady-state wet length between the droplethe wall on the Bond number. Other parameters are the same as in Fig

FIG. 11. Dependency of the steady-state contact-line velocity on the Bnumber. Other parameters are the same as in Fig. 10.


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tween the droplet and the displacing fluid are both 1. Asarea of the droplet increases, the critical Bond numbercreases monotonically. It is helpful to study the effect ofA onthe critical capillary number, which is the ratio of viscoussurface forces and defined asCa5Um1 /s by Schleizer andBonnecaze.19 Noticing here thatr15r2 and m15m2 , wehaveCa5Um1 /s5r2gh2/s. The only difference betweenthe definition ofCa andB0 is thath2, the square of the widthof the channel, is used in the definition ofCa , while A, thearea of the static droplet, is used in the definition ofB0 .Figure 14 shows the dependency of the critical capillnumber on the dimensionless area of the droplet. It is cthat the critical capillary number decreases monotonicallythe static droplet area increases, which is intuitively corrand confirms the conclusion of Schleizer and Bonnecazethe droplet area increases, the relative velocity betweendroplet and the displacing fluid increases and so do the

FIG. 12. The steady-state shape of the droplet at different Bond numOther parameters are the same as in Fig. 2:~a! u578°, ~b! u590°, ~c! u5118°.


e-

yarsts

hes-

cous forces between them. As a result, all or part ofdroplet is able to detach from the wall or pinch off from threst of it at a smaller body force strength.

4. Effects of density and viscosity ratios

We also performed the simulations of other cases wvarious density and viscosity ratios. The results are sumrized in Table II. The dimensionless droplet area is 0.Increasing the viscosity of the droplet increases the viscforces between the droplet and the displacing fluid, ahence decreases the critical Bond number. Increasingdensity of the droplet, however, results in a higher criticBond number. The reason is understandable: Displacem

rs.

FIG. 13. The dependency of the critical Bond number on the dimensionarea of the static droplet, withr2 /r151, n2 /n151, g125g2150.2, andcontact anglesu578°, u590°, andu5118°.

FIG. 14. The dependency of the critical capillary number on the dimensless area of the static droplet, withr2 /r151, n2 /n151, g125g2150.2, andcontact anglesu578°, u590°, andu5118°.


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of a heavy fluid by a light one is more difficult than othewise, provided other conditions are the same.

IV. CONCLUSIONS

We have simulated the displacement of a twdimensional immiscible droplet of various wettabilities inchannel, subject to gravitational forces by the LB methWe found that at a very small Bond number, the droplet wdifferent contact angles behaved similarly, the wet lenbetween the droplet and the wall decreased with time unsteady shape was reached, which was different from themerical results of Schleizer and Bonnecaze.19 Their simula-tions showed that the wetting droplet spread along the waany capillary number. One possible reason of the discrepamight be due to the presence of contact angle hysteresour simulations, since we made no assumption about thelationship between the contact angle and the velocity ofcontact line. Hence, the advancing and receding conangles were different from the static one in our study.indicated in their paper, including the effects of contact anhysteresis might significantly change the results. Our simlation results showed that these effects were more imporat very small Bond number because when the dropletvery slowly, the contact line yielded first would have a mopronounced effect on the behavior of the droplet.

As the Bond number increased, we found that the tievolution curves of the dimensionless wet length at differcontact angles separated from each other. There also exa critical Bond number, above which no steadily slididroplet would be observed and the entire droplet wouldtach from the wall, or the upper portion of the droplet woupinch off from the lower portion of it, depending on its statcontact angle. Increasing the contact angle resulted ilarger fraction of droplet being entrained in the bulk. Oobservations are in line with the simulation results andassertion by Schleizer and Bonnecaze.19 We also found as ageneral trend that the critical Bond number decreased ascontact angle increased.

In the nonwetting droplet displacement, the steady-swet length decreased monotonically and the steady-state

TABLE II. Effect of viscosity and density ratios on the critical Bond number.

u ~deg! n2 /n1 r2 /r1 B0c

78 1 1 0.859378 0.5 1 0.984778 2 1 0.538978 1 2 1.352778 1 0.5 0.467990 1 1 0.603790 0.5 1 0.720590 2 1 0.436490 1 2 1.021790 1 0.5 0.3849118 1 1 0.2439118 0.5 1 0.3095118 2 1 0.2127118 1 2 0.4009118 1 0.5 0.1993


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locity of the contact lines increased linearly as the Bonumber increased. For the case of a contact angle less thequal to 90°, the situation was more complicated. Tsteady-state wet length decreased first and then increasthe Bond number increased. Whenu590°, the steady-statevelocity of the contact lines also increased linearly with tincrease of the Bond number but with a smaller slope thwhen u5118°. When u578°, however, the steady-stacontact-line velocity varied linearly in two regions. The trasition between the two regions corresponded to the regwhere there was a great change in the droplet shape.

We also performed simulations with various dropsizes, density ratios, and viscosity ratios and found that wthe increase of the viscosity ratio, or with the decrease ofdensity ratio, the critical Bond number decreased; withincrease of the droplet size, the critical Bond numbercreased, and the critical capillary number decreased.

The LB method used here is not limited to simulating tdisplacement of an immiscible droplet in a channel with flsolid boundaries. It may also be effective in simulating tdisplacement of partially miscible fluids in complex geometries, which is the topic of our future work.

ACKNOWLEDGMENTS

This work was partially funded by the NSF Grant N0103408 to The Johns Hopkins University and by tLDRD/ER Project No. 99025 from Los Alamos NationLaboratory, which is operated by the University of Californfor the U.S. Department of Energy.

1R. G. Cox, ‘‘The dynamics of the spreading of liquids on a solid surfacJ. Fluid Mech.168, 169 ~1986!.

2H. Zhou and C. Pozrikidis, ‘‘Pressure-driven flow of suspensions of liqdrops,’’ Phys. Fluids6, 80 ~1994!.

3A. Borhan and C. F. Mao, ‘‘Effect of surfactants on the motion of drothrough circular tubes,’’ Phys. Fluids A4, 2628~1992!.

4T. M. Tsai and M. J. Miksis, ‘‘Dynamics of a drop in a constricted caplary tube,’’ J. Fluid Mech.274, 197 ~1994!.

5E. B. Dussan, ‘‘On the ability of drops to stick to surfaces of solids,’’Fluid Mech.174, 381 ~1987!.

6X. Li and C. Pozrikidis, ‘‘Shear flow over a liquid drop adhering to a sosurface,’’ J. Fluid Mech.307, 167 ~1996!.

7J. Q. Feng and O. A. Basaran, ‘‘Shear flow over a translationally cylincal bubble pinned on a slot in a plane wall,’’ J. Fluid Mech.275, 351~1994!.

8P. Dimitrakopoulos and J. J. L. Higdon, ‘‘Displacement of fluid droplefrom solid surfaces in low-Reynolds-number shear flows,’’ J. Fluid Me336, 351 ~1997!.

9P. Dimitrakopoulos and J. J. L. Higdon, ‘‘On the displacement of thrdimensional fluid droplets adhering to a plane wall in viscous pressdriven flows,’’ J. Fluid Mech.435, 327 ~2001!.

10E. B. Dussan, ‘‘The moving contact line: the slip boundary condition,’’Fluid Mech.77, 665 ~1976!.

11L. M. Hocking, ‘‘A moving fluid interface on a rough surface,’’ J. FluidMech.76, 801 ~1976!.

12K. M. Jansons, ‘‘Moving contact lines on a two-dimensional rough sface,’’ J. Fluid Mech.154, 1 ~1985!.

13M. Y. Zhou and P. Sheng, ‘‘Dynamics of immiscible fluid displacementa capillary tube,’’ Phys. Rev. Lett.64, 882 ~1990!.

14J. Koplik, J. R. Banavar, and J. F. Willemsen, ‘‘Molecular dynamics oPoiseuille flow and moving contact lines,’’ Phys. Rev. Lett.60, 1282~1988!.

15Y. D. Shikhmurzaev, ‘‘Moving contact lines in liquid/liquid/solid systems,’’ J. Fluid Mech.334, 211 ~1997!.

16L. Fan, H. Fang, and Z. Lin, ‘‘Simulation of contact line dynamics in


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ingnn

dy

o-ur

da

a

,’’

n—

tio

zau

ltz

ihy

o

ice

ul-

s

a

z-

el

o

of

nn

,’’la-

m-h–

of

idm-

-d,’’

nget.

nal

kes

,’’Uni-

ofog.

s,’’

alann


two-dimensional capillary tube by the lattice Boltzmann model,’’ PhRev. E63, 051603~2001!.

17P. Raiskinmaki, A. Koponen, J. Merikoski, and J. Timonen, ‘‘Spreaddynamics of three-dimensional droplets by the lattice-Boltzmamethod,’’ Comput. Mater. Sci.18, 7 ~2000!.

18D. Grubert and J. M. Yeomans, ‘‘Mesoscale modeling of contact linenamics,’’ Comput. Phys. Commun.121–122, 236 ~1999!.

19A. D. Schleizer and R. T. Bonnecaze, ‘‘Displacement of a twdimensional immiscible droplet adhering to a wall in shear and pressdriven flows,’’ J. Fluid Mech.383, 29 ~1998!.

20F. J. Alexander, S. Chen, and D. W. Grunau, ‘‘Hydrodynamics spinodecomposition: Growth kinetics and scaling,’’ Phys. Rev. B48, 634~1993!.

21M. Cieplak, ‘‘Rupture and coalescence in 2-dimensional cellular-automfluids,’’ Phys. Rev. E51, 4353~1995!.

22S. Chen and G. D. Doolen, ‘‘Lattice Boltzmann method for fluid flowsAnnu. Rev. Fluid Mech.30, 329 ~1998!.

23R. Benzi, S. Succi, and M. Vergassola, ‘‘The lattice Boltzmann-equatioTheory and applications,’’ Phys. Rep.222, 145 ~1992!.

24D. H. Rothman and S. Zaleski, ‘‘Lattice-gas models of phase separaInterfaces, phase transitions and multiphase flow,’’ Rev. Mod. Phys.66,1417 ~1994!.

25S. Chen, S. P. Dawson, G. D. Goolen, D. R. Janecky, and A. Lawnic‘‘Lattice methods and their applications to reacting systems,’’ CompChem. Eng.19, 617 ~1995!.

26Y. Qian, S. Succi, and S. A. Orszag, ‘‘Recent advances in lattice Bomann computing,’’ Annu. Rev. Comput. Phys.3, 195 ~1995!.

27D. Zhang, R. Zhang, S. Chen, and W. E. Soll, ‘‘Pore scale study of flowporous media: Scale dependency, REV, and statistical REV,’’ GeopRes. Lett.27, 1195~2000!.

28Q. Kang, D. Zhang, S. Chen, and X. He, ‘‘Lattice Boltzmann simulationchemical dissolution in porous media,’’ Phys. Rev. E65, 036318~2002!.

29A. K. Gunstensen, D. H. Rothman, S. Zaleski, and G. Zanetti, ‘‘LattBoltzmann model of immiscible fluids,’’ Phys. Rev. A43, 4320~1991!.

30D. Grunau, S. Chen, and K. Eggert, ‘‘A lattice Boltzmann model for mtiphase fluid flows,’’ Phys. Fluids A5, 2557~1993!.

31X. Shan and H. Chen, ‘‘Lattice Boltzmann model for simulation flowwith multiple phases and components,’’ Phys. Rev. E47, 1815~1993!.

32X. Shan and H. Chen, ‘‘Simulation of nonideal gases and liquid-gas phtransitions by the lattice Boltzmann equation,’’ Phys. Rev. E49, 2941~1994!.

33X. Shan and G. D. Doolen, ‘‘Diffusion in a multicomponent lattice Boltmann equation model,’’ Phys. Rev. E54, 3614~1996!.

34X. Shan and G. D. Doolen, ‘‘Multicomponent lattice-Boltzmann modwith interparticle interaction,’’ J. Stat. Phys.81, 379 ~1995!.

35S. Hou, X. Shan, Q. Zou, G. Doolen, and W. Soll, ‘‘Evaluation of tw


.

-

e-

l

ta

n:

k,t.

-

ns.

f

se

lattice Boltzmann models for multiphase flows,’’ J. Comput. Phys.138,695 ~1997!.

36M. Swift, W. Osborn, and J. Yeomans, ‘‘Lattice Boltzmann simulationnonideal fluids,’’ Phys. Rev. Lett.75, 830 ~1995!.

37M. Swift, S. Orlandini, W. Osborn, and J. Yeomans, ‘‘Lattice Boltzmasimulations of liquid-gas and binary-fluid systems,’’ Phys. Rev. E54, 5041~1996!.

38R. Zhang, ‘‘Lattice Boltzmann approach for immiscible multiphase flowPh.D. thesis, Department of Mechanical Engineering, University of Deware, 1999.

39X. He, S. Chen, and R. Zhang, ‘‘A lattice Boltzmann scheme for incopressible multiphase flow and its application in simulation of RayleigTaylor instability,’’ J. Comput. Phys.152, 642 ~1999!.

40A. K. Gunstensen and D. H. Rothman, ‘‘Lattice-Boltzmann studiesimmiscible two-phase flow through porous media,’’ J. Geophys. Res.98,6431 ~1993!.

41S. Chen, G. D. Doolen, and K. G. Eggert, ‘‘Lattice-Boltzmann fludynamics—A versatile tool for multiphase fluid dynamics and other coplicated flows,’’ Los Alamos Sci.22, 98 ~1994!.

42N. S. Martys and H. Chen, ‘‘Simulation of multicomponent fluids in complex three-dimensional geometries by the lattice Boltzmann methoPhys. Rev. E53, 743 ~1996!.

43K. Langaas and D. Grubert, ‘‘Lattice Boltzmann simulations of wettiand its application to disproportionate permeability reducing gel,’’ J. PSci. Eng.24, 199 ~1999!.

44X. He, R. Zhang, S. Chen, and G. D. Doolen, ‘‘On three-dimensioRayleigh–Taylor instability,’’ Phys. Fluids11, 1143~1999!.

45H. Chen, S. Chen, and W. H. Matthaeus, ‘‘Recovery of the Navier–Stoequations using a lattice-gas Boltzmann method,’’ Phys. Rev. A45, R5339~1992!.

46S. Hou, ‘‘Lattice Boltzmann method for incompressible, viscous flowPh.D. thesis, Department of Mechanical Engineering, Kansas Stateversity, 1995.

47B. R. Sehgal, R. R. Nourgaliev, and T. N. Dinh, ‘‘Numerical simulationdroplet deformation and break-up by lattice Boltzmann method,’’ PrNucl. Energy34, 471 ~1999!.

48F. A. L. Dullien, Porous Media: Fluid Transport and Pore Structure~Aca-demic, New York, 1992!.

49S. Wolfram, ‘‘Cellular automaton fluids. 1: Basic theory,’’ J. Stat. Phys.45,471 ~1986!.

50P. Lavallee, J. P. Boon, and A. Noullez, ‘‘Boundaries in lattice gas flowPhysica D47, 233 ~1991!.

51Z. L. Yang, T. N. Dinh, R. R. Nourgaliev, and B. R. Sehgal, ‘‘Numericinvestigation of bubble growth and detachment by the lattice-Boltzmmethod,’’ Int. J. Heat Mass Transf.44, 195 ~2001!.


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