Prasanna R. Redapangu

Kirti Chandra Sahue-mail: ksahu@iith.ac.in

Department of Chemical Engineering,

Indian Institute of Technology Hyderabad,

Ordinance Factory Estate, Yeddumailaram,

Andhra Pradesh, Hyderabad 502205, India

S. P. VankaDepartment of Chemical Engineering,

Indian Institute of Technology Hyderabad,

Ordinance Factory Estate, Yeddumailaram,

Andhra Pradesh, Hyderabad 502205, India;

Department of Mechanical

Science and Engineering,

University of Illinois at Urbana-Champaign,

Urbana, Il 61822

A Lattice Boltzmann Simulationof Three-DimensionalDisplacement Flowof Two ImmiscibleLiquids in a Square DuctA three-dimensional, multiphase lattice Boltzmann approach is used to study a pressure-driven displacement flow of two immiscible liquids of different densities and viscosities ina square duct. A three-dimensional, 15-velocity (D3Q15) lattice model is used. Theeffects of channel inclination, viscosity, and density contrasts are investigated. The con-tours of the density and the average viscosity profiles in different planes are plotted andcompared with those obtained in a two-dimensional channel. We demonstrate thatthe flow dynamics in a three-dimensional channel is quite different as compared to thatof a two-dimensional channel. We found that the flow is relatively more coherent in athree-dimensional channel than that in a two-dimensional channel. A new screw-typeinstability is seen in the three-dimensional channel that cannot be observed in thetwo-dimensional channel. [DOI: 10.1115/1.4024998]

1 Introduction

The displacement of one fluid by another under the action of apressure gradient is known as displacement flow. This paperinvestigates the three-dimensional displacement flow of a liquidoriginally occupying a square duct by another immiscible liquidof different density and viscosity, which is injected from the inletof the channel. The work is an extension of our previous study, inwhich the displacement flow in a 2D channel was simulated [1].The displacement flow of one fluid by another can be found inmany industrial and geological applications, such as transportationof crude oil in pipelines [2], oil recovery, food processing, coat-ing, flow of lava inside the earth [3], etc. In the latter case, thetemperature gradient also plays an important role in the flow dy-namics. In porous media or in a Hele–Shaw cell, the displacementof a highly viscous fluid by a less viscous one becomes unstable,forming various instability patterns, known as viscous fingering[4].

Due to their relevance to practical applications, displacementflow has been studied by considering miscible [5–15] as well asimmiscible fluids [1,16–20]. In a miscible system, Goyal and Mei-burg [7] numerically studied the displacement of a larger viscosityfluid by a less viscous one in a Hele–Shaw cell and found qualita-tively similar flow fields observed in the experiment of Petitjeansand Maxworthy [8]. These results are also in good agreement withthe theoretical predictions of Lajeunesse et al. [21]. Theyobserved that the two-dimensional instability patterns become 3Dat higher flow rates. Taghavi et al. [11,12] studied the displace-ment flow of two miscible liquids in a system having significantbuoyancy effects. They observed Kelvin–Helmholtz–like instabil-ities at low imposed velocities in the exchange flow dominated re-gime. Sahu et al. [9,10] studied the pressure-driven flow of twomiscible liquids of different densities and viscosities in aninclined channel. They investigated the effects of Reynolds num-ber, Schmidt number, Froude number, and angle of inclination.They also studied the effects of an imposed infinitesimal disturb-

ance on the flow by conducting a linear stability analysis. Theeffects of temperature gradient and wall heating on a similar sys-tem were included in Sahu et al. [22].

The flow characteristics in the above-mentioned systems arevery complex, even without the presence of a sharp interface,where one could have a sudden change in the fluid properties. Of-ten the traditional computational methods, which solve theNavier–Stokes equation, are used to study these problems. Immis-cible systems are expensive computationally, due to the interfacialdynamics, which happen in a very narrow region. A big task whilehandling such systems is to track the dynamically changing inter-face, which requires additional computational effort. Due to this,a very few computational studies have been carried out on immis-cible systems. The instabilities in immiscible systems have beenstudied mainly in core-annular/three-layer/two-layer configurationby several researchers (see the extensive review by Joseph et al.[2] and a recent review on viscosity stratified flows by Govindara-jan and Sahu [23]). Considering a two-layer Couette/Poiseuilleflow, Yih [24] was the first to show that long waves on a viscosityinterface can be unstable at any Reynolds number. Since then,several researchers (for example, Refs. [25–27]) have studied theeffects of viscosity ratio, density ratio, and height of the interfacein such systems and have investigated both the long- and short-wave instabilities by conducting linear stability analyses.

During the past few decades, lattice Boltzmann method (LBM)has emerged as a promising alternative technique for fluid flowsimulations [28]. There are mainly four distinctly different LBMapproaches for multiphase flows: the color segregation method ofGunstensen et al. [29], the method of Shan and Chen [30], the freeenergy approach of Swift et al. [31], and the method of He and co-workers [32–34]. These methods have been previously used by afew researchers [1,17,20] to study the displacement flows of twoimmiscible liquids. The effects of viscosity ratio, surface tension,wettability, and capillary number was discussed by Chin et al.[18], Grosfils et al. [19], Kang et al. [17], and Dong et al. [20],respectively. Due to the low characteristic Reynolds number con-sidered, these studies did not observe any interfacial instabilities.Using a LBM approach, Langaas and Yeomans [35] studied thefingering phenomenon of two immiscible fluids with different vis-cosities and investigated the effects of surface tension on the flow

Contributed by the Fluids Engineering Division of ASME for publication in theJOURNAL OF FLUIDS ENGINEERING. Manuscript received March 5, 2013; finalmanuscript received July 10, 2013; published online September 19, 2013. Assoc.Editor: Samuel Paolucci.

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dynamics. Recently, Redapangu et al. [1] revisited the displace-ment flow of one liquid by another immiscible liquid in atwo-dimensional channel using the two-dimensional version of atwo-phase LBM based on the Zhang et al. [32] approach.

In spite of the large number of studies carried out on displace-ment flows, to the best of our knowledge, none of them haveexamined the pressure-driven displacement flow of immisciblefluids in three-dimensional duct at moderate-to-large Reynoldsnumbers. A three-dimensional study is necessary in order tounderstand the complete picture of the instabilities. Riaz and Mei-burg [36] and Oliveira and Meiburg [37] numerically studiedthree-dimensional displacement flow of two miscible fluids in po-rous media and Hele–Shaw cell, respectively. They found thatsome of the flow features in 3D are qualitatively different fromthose obtained in 2D simulation. Similarly, Hallez and Magnaudet[38] investigated the effects of buoyancy in inclined channel/pipeand found that the vortical structures are more coherent in two-dimensional geometry than those in three-dimensional geometry.

In the present work, we study the three-dimensional pressure-driven displacement flow of two immiscible liquids of differentdensities and viscosities in a square duct using a multiphase latticeBoltzmann method. To achieve high computational efficiency, weimplemented our LBM algorithm on a graphics processing unit(GPU) [39]. Our present GPU-based LBM solver is 25 timesfaster than the corresponding central processing unit–based codeon a single core. We do not describe the GPU implementation inthis paper, as it has been discussed in our previous papers [39,40].However, it is worth mentioning here that GPU-based simulationwas very beneficial for speeding up the computations. In this pa-per, we have investigated the effects of channel inclination, vis-cosity, and density contrasts. The flow behavior observed in 3D iscompared with that obtained in a 2D channel.

The rest of the paper is organized as follows. Details of theproblem formulation are given in Sec. 2, the LBM approach usedto carry out the computations is discussed in Sec. 3, the results ofthis study are provided in Sec. 4, and concluding remarks aregiven in Sec. 5.

2 Formulation

We consider the pressure-driven displacement of one liquid(“liquid 2”: viscosity l2 and density q2) by an another liquid(“liquid 1”: viscosity l1 and density q1). The liquids are immisci-ble and are assumed to be incompressible Newtonian. The sche-matic of the geometry is shown in Fig. 1. A three-dimensionalrectangular coordinate system, (x, y, z), is used to model the flowdynamics where x, y, and z denote the axial, vertical, and the azi-

muthal coordinates, respectively. The inlet and the outlet of thechannel are located at x¼ 0 and L, respectively. The rigid andimpermeable walls of the channel are located at y¼ 0, H andz¼ 0, W. Here, we consider W¼H, and the aspect ratio of channelL/H is 32. h is the angle of inclination measured with the horizon-tal. g is the acceleration due to gravity. The components of gravitygxð� gsinhÞ and gyð� gcoshÞ act in the negative axial and nega-tive vertical directions, respectively. The gravity component inthe azimuthal direction, gz, equals to zero. In the initial configura-tion, the channel portions from 0� x�H/4 and H/4� x� L arefilled with liquid 1 and liquid 2, respectively.

At t¼ 0, the liquids inside the channel are stationary when afully developed flow due to a pressure gradient is imposed at theinlet. This is given by

@2u

@y2þ @

2u

@z2¼ Re

dp

dx(1)

where u is the velocity component in the axial direction and p isthe pressure. The velocity components in the vertical (v) and azi-muthal (w) directions are zero at the inlet. The volumetric flowrate, Q, is maintained constant for all the simulations performedin the present study, and the imposed pressure-gradient is calcu-lated from the prescribed volumetric flow rate. The no-slip andno-penetration boundary conditions are imposed at the walls, andthe Neumann condition is prescribed at the outlet (x¼ L). Wedefine the Reynolds number based on the viscosity of liquid 1,Re � Qq1=Wl1. Viscosity ratio, m, is defined as the ratio of vis-cosity of liquid 2 to the viscosity of liquid 1, (i.e., m � l2=l1).For a value of m above 1, the displacing liquid is less viscous thanthe liquid inside the channel and vice versa for m below 1. Thedensity contrasts between the liquids are measured by a dimen-sionless quantity, Atwood number At � ðq2 � q1Þ=ðq2 þ q1Þð Þ;thus, At> 0 indicates a lighter liquid entering the channel. Thegravity effects can be characterized by Froude number (Fr),defined as Fr � Q=HW

ffiffiffiffiffiffiffiffiffiffiffiAtgHp

. The magnitude of surface tensionis measured by j, which is related to surface tension, r (please seeSec. 3). The surface tension is characterized by capillary number,Cað� Ql1=rHÞ. The dimensionless time scale is represented astð� H2W=QÞ.

3 Numerical Method

The simulations are performed with the two-phase lattice Boltz-mann method (LBM) previously proposed by He and coworkers[32] and used in recent study of Sahu and Vanka [41]. A three-dimensional, 15-velocity lattice model (D3Q15) is used. TheLBM simulates fluid flow problems using discrete density distri-bution functions. In this method, two distribution functions (indexdistribution function (f) and the pressure distribution function(g))are used to track the interface and to calculate the macroscopicproperties of the fluids. The evolution equations of distributionfunctions are given by

f aðxþ eadt; tþdtÞ� f aðx; tÞ¼�f aðx; tÞ� f eq

a ðx; tÞs

�2s�1

2sðea�uÞ �rwðuÞ

c2s

CaðuÞdt

(2)

gaðxþ eadt; tþ dtÞ � gaðx; tÞ ¼ �gaðx; tÞ � geq

a ðx; tÞs

þ 2s� 1

2sðea � uÞ �

hCaðuÞ Fs þGð Þ � CaðuÞð

�Cað0ÞÞrwðqÞidt (3)

Here, u¼ (u, v, w) represents the three-dimensional velocity field,dt is the time step, a is the lattice direction (shown in Fig. 2), and

Fig. 1 Schematic describing the geometry and the initialconfiguration

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s is the relaxation time. The kinematic viscosity, �, is related tothe relaxation time by the expression � ¼ ðs� 1=2Þdtc2

s .In D3Q15 lattice, lattice velocities and the weighing coeffi-

cients are given by

e1; e2; :::; e15½ � ¼

0 1 �1 0 0 0 0 1 �1 1 �1 1 �1 1 �1

0 0 0 1 �1 0 0 1 1 �1 �1 1 1 �1 �1

0 0 0 0 0 1 �1 1 1 1 1 �1 �1 �1 �1

2664

3775 (4)

t1; t2; :::; t15½ � ¼ 2

9;1

9;1

9;1

9;1

9;1

9;1

9;

1

72;

1

72;

1

72;

1

72;

1

72;

1

72;

1

72;

1

72

� �(5)

respectively. Ca(u) is a function of macroscopic velocity u, whichis given by

CaðuÞ ¼ ta 1þ ea � uc2

s

þ ðea � uÞ2

2c4s

� u2

2c2s

" #(6)

The equilibrium distribution functions, f eqa and geq

a , are given by

f eqa ¼ ta/ 1þ ea � u

c2s

þ ðea � uÞ2

2c4s

� u2

2c2s

" #(7)

geqa ¼ ta pþ qc2

s

ea � uc2

s

þ ðea � uÞ2

2c4s

� u2

2c2s

!" #(8)

where c2s ¼ 1=3.

The term rwð/Þ mimics the intermolecular interactions fornonideal gases or dense fluids, which plays an important role inphase segregation. Thus, it needs to be computed accurately; afourth order compact scheme is used to discretize rwð/Þ andrwðqÞ [42]. We use the Carnahan–Starling equation of state forw(/), which describes nonideal gases and fluids [43,44],

wð/Þ ¼ c2s /

1þ /þ /2 � /3

ð1� /Þ3� 1

" #� a/2 (9)

where a determines the strength of molecular interactions. Thecritical value of Carnahan–Starling equation of state isac¼ 3.53374. Choosing a value of a> ac, one could capture phaseseparation of immiscible liquids; however, selecting a very largea value can create numerical error [32]. Therefore, we chose a¼ 4in the present study. The term w(q) is given by p� qc2

s . The termsFs and G in Eq. (3) represent the surface tension force and thegravity forces, respectively,

Fs ¼ j/rr2/ and G ¼ ðq� qmÞg (10)

where j is the magnitude of surface tension andqm � ðq1 þ q2Þ=2.

The surface tension, r, can be related to j as follows [45]:

r ¼ jð@/@f

� �2

df ¼ jIðaÞ (11)

where f is the direction normal to the interface. Zhang et al. [32]obtained an analytical expression for I(a) to calculate surface ten-sion, which is given by

IðaÞ ¼ 0:1518ða� acÞ1:5

1þ 3:385ða� acÞ0:5(12)

The index function (/), pressure (p), and the velocity field (u)are calculated from the distribution functions as

/ ¼X

f a (13)

p ¼X

g

a� 1

2u � rwðqÞdt (14)

quc2s ¼

Xe

aga þc2

s

2ðFs þGÞdt (15)

The liquid density and kinematic viscosity can be estimated fromthe index function using the following equations:

qð/Þ ¼ q2 þ/� /2

/1 � /2

ðq1 � q2Þ (16)

�ð/Þ ¼ �2 þ/� /2

/1 � /2

ð�1 � �2Þ (17)

Here, �1 and �2 are the kinematic viscosities of liquids 1 and 2,respectively. /1 and /2 are minimum and maximum values of theindex function; in the present study, /1 and /2 are given values of0.02381 and 0.2508, respectively [32]. In Sec. 4, we discuss theresults obtained from three-dimensional simulations followed bycomparison of the results with those obtained from two-dimensional simulations.Fig. 2 Schematic diagram of D3Q15 lattice

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4 Results and Discussion

We start presenting our results by first showing the results ofgrid independency tests. The temporal variation of the dimension-less mass of liquid 2, Mt/M0, is plotted for different grids in Fig. 3.Here, Mt represents the instantaneous mass of liquid 2 and isdefined as Mt¼q2

ÐW0

ÐH0

Ð L0ð/�/1Þ=ð/2�/1Þdxdydz. M0ð�q2ð/

�/1Þ=ð/2�/1ÞLHWÞ is the mass of liquid 2 initially occupyingthe channel at t¼0. The parameters used to generate this plot areRe¼100, m¼10, At¼0.2, Fr¼5, Ca¼29.9, and h¼45 deg. InFig. 3, it can be seen that Mt/M0 decrease monotonically with timeas liquid 2 is displaced by liquid 1. The dotted line represents theanalytical expression for plug-flow displacement (given byMt=M0¼1� tH=L). It can be seen that, during the early times(t<10), the displacement rate is close to that of the analytical so-lution. However, at latter times, the actual displacement is slowerthan that of the plug-flow displacement process. Similar behaviorwas observed in the two-dimensional simulation of Redapanguet al. [1], but for miscible fluids, Sahu et al. [10] found that thedisplacement rate is much faster than that of the plug-flow dis-placement. Redapangu et al. [1] attributed this difference to thephase change associated with the miscible systems. In Fig. 3, itcan be seen that the results obtained using different grids arealmost the same (with a maximum absolute error less than 1%).The contours of / in the x-y plane obtained from the three-dimensional simulation for the parameter values the same asFig. 3 are shown in Fig. 4 for t¼10 and 30 for different grids con-sidered. It can be seen that the flow dynamics are qualitativelysimilar for all the grids considered; however, close inspection ofthese results reveals that there is a slight increase in the velocityof the finger tip for the coarsest grid, which tends to converge aswe increase the grid points. Therefore, we use a 2112�66�66grid for the rest of the simulations.

The spatiotemporal evolution of the isosurface of / in the range0.023–0.25 (which represents the interface between the fluids) isshown in Fig. 5 for the parameter values the same as those used togenerate Fig. 3. This set of parameter values represents a situationwhen a less viscous and less dense liquid displaces a highly vis-cous and highly dense liquid. In this system, due to the imposedpressure gradient at the inlet, a “finger” of liquid 1 penetrates intothe region of liquid 2. As the channel is inclined at an angleh¼ 45 deg, the motion induced by the imposed pressure gradientis opposed by the flow, due to the gravitational force (proportionalto gx) in the negative axial direction. Therefore, the flow dynamicsare due to the competition between the imposed pressure gradientand the gravity force. At the same time, the component of thegravitational force in the vertical direction (proportional to gy)acts to segregate the two liquids. The finger becomes asymmetri-cal because of this force. We observed that, for t� 10, the inter-face separating the two liquids becomes unstable, formingnonlinear interfacial waves that grow in time. At later times(t> 18 for this set of parameter values), the instabilities of cork-screw pattern are observed. This pattern is the consequence of theYih-type instabilities [24] in the linear regime. By conducting alinear stability analysis in a miscible core-annular flow configura-tion, Selvam et al. [46] found that, at low diffusivity, the cork-screw mode becomes more unstable than the axisymmetric mode.The corkscrew-type pattern was also observed by Scoffoni et al.[47] in miscible displacement flow in a vertical pipe.

Next, we discuss the effects of angles of inclination, h, by plot-ting the contours of / at t¼ 20 in the x-y plane at z¼W/2 and inx-z plane at y¼H/2 in Figs. 6(a) and 6(b), respectively. The restof the parameter values are Re¼ 100, At¼ 0.2, m¼ 10, Fr¼ 1,and Ca¼ 29.9. The plots in panel (a) of this figure are the viewsin the z direction. It can be seen that the intensity of the instabil-ities increases with increasing angles of inclination. Also, due tothe decrease in the gravitational force in the y direction, for higherangles of inclination, core-annular structures form as the finger

Fig. 3 Variation of mass fraction of the displaced liquid (Mt/M0)with time for different grids. The rest of the parameters areRe 5 100, m 5 10, At 5 0.2, Fr 5 5, Ca 5 29.9, and h 5 45 deg. Thedotted line represents the analytical solution of the plug-flowdisplacement, given by Mt=M0 5 1� tH=L.

Fig. 4 Contours of the index function, /, at (a) t 5 10 and (b) t 5 30 in the x-y plane at z 5 W/2 fordifferent grids. The parameters are the same as those of Fig. 3.

Fig. 5 Evolution of the isosurface of / at the interface at differ-ent times (from left to right: t 5 12, 18, 30, and 50) for the simula-tion domain of 2112 3 66 3 66 grid. The parameters areRe 5 100, m 5 10, At 5 0.2, Fr 5 5, Ca 5 29.9, and h 5 45 deg. Theflow is in the positive x direction.

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penetrates downstream. However, the gravitational force in thenegative axial direction increases with increasing h, which makesthe trailing fingers asymmetrical in the upstream regime. Weobserve that the fingers have a blunt “nose,” separating the twofluids in these cases. For the largest angle of inclination consid-ered (h¼ 85 deg), the droplet of fluid 1 is detached from the mainfinger at t¼ 20. As the gravitational force in the y directiondecreases with increase in h, for higher angles of inclination, it isdominated by the effects created due to the imposed pressure gra-dient, therefore leaving patches of the heavier fluid near the topwall at y¼H. This is shown in Fig. 6(b), when one looks from thetop in the y direction.

The contours of / in the y-z plane at x¼ L/2 are shown in Fig. 7for different angles of inclination. The parameter values are thesame as those used to generate Fig. 6. This is a view of a crosssection at the midlength of the channel in the x direction. Forsmaller angles of inclination, the gravitational force in the nega-tive y direction is stronger, which tries to settle the heavier fluid inthe bottom part of the channel. This can be observed for h� 30deg in Fig. 7. However, as discussed above for higher angles of in-clination, the pressure gradient dominates the flow dynamics, cre-

ating a core-annular configuration, as shown in Fig. 7 for h> 60deg.

In Figs. 8(a) and 8(b), the axial variation of normalized averageviscosity, �lyzð� lyz=l

0yzÞ, is plotted at different times for h¼ 30

deg and h¼ 85 deg, respectively. Here, lyz¼ð1=HWÞÐW

0

ÐH0

ldydzand l0

yz is the value of lyz at x¼0. The rest of the parameters arethe same as those in Fig. 6. It can be seen that, as expected fromthe discussion of Fig. 6(a), the variation becomes increasinglycomplex with increase in h. Similarly, the vertical variation ofnormalized average viscosity, �lxz �lxz=l

0xz

� �, where lxz¼ð1=LWÞÐW

0

Ð L0

ldxdz, is plotted in Fig. 9 for different times. The value of lxz

at y¼0 is designated by l0xz. It can be seen that, for higher angles of

inclination, a core-annular structure is formed, which was also dis-cussed in Fig. 7. On the other hand, for lower angles of inclination,a two-layer structure is formed as the heavier fluid settles in thelower part of the channel.

Next, we compare the three-dimensional results with thoseobtained from the two-dimensional simulations. For this, we plotthe axial and the vertical variation of �lyð� ly=l

0yÞ and

�lxð� lx=l0xÞ in Figs. 10 and 11, respectively, for h¼ 5 deg and

h¼ 85 deg. Here, ly ¼ 1=HÐH

0ldy and lx ¼ 1=L

Ð L0

ldx. l0y and

Fig. 6 Contours of the index function, /, at t 5 20 in (a) x-y plane at z 5 W/2 and (b) x-z plane aty 5 H/2. The parameters are Re 5 100, At 5 0.2, m 5 10, Fr 5 1, and Ca 5 29.9.

Fig. 7 Contours of the index function, /, at t 5 20 in the y-z plane at x 5 L/2. Theparameters are Re 5 100, At 5 0.2, m 5 10, Fr 5 1, and Ca 5 29.9.

Fig. 8 Axial variation of normalized average viscosity, �lyz ð� lyz=l0yz Þ for (a) h 5 30 deg and (b)

h 5 85 deg. The rest of the parameters are Re 5 100, At 5 0.2, m 5 10, Fr 5 1, and Ca 5 29.9. Here,lyz ¼ ð1=HW Þ

RW

0

RH

0 ldydz and l0yz is the value of lyz at x 5 0.

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l0x are the values of ly at x¼ 0 and lx at y¼ 0, respectively. It can

be seen in Fig. 10 that the axial variation of �ly is more chaotic inthe two-dimensional channel as compared to that in three-dimensional channel (shown in Fig. 8). This is due to the stabiliz-ing effects of the azimuthal walls to the disturbances of smallerwavelength. This behavior is similar to the surface-tension effectsin two-dimensional channel [48]. Inspection of the vertical varia-tion of �lx reveals that the two-dimensional flow remains core an-nular, even for small angles of inclination. This is in contrastwith the phenomena observed in three-dimensional computations(Fig. 9).

Then, we compare the results obtained from the two- and three-dimensional channels by plotting the spatiotemporal evolution of/ contours in Figs. 12(a) and 12(b), respectively, for Re¼ 100,m¼ 20, At¼ 0.2, Fr¼ 5, h¼ 30 deg, and Ca¼ 29.9. It can beobserved that 3D effect reduces the short-wave instabilities, whichare apparent in the two-dimensional channel at later times (seenear the exit of the channel at t� 24 for this set of parameters).This behavior is also discussed above. Close inspection of Fig. 12also reveals that the velocity of the fingertip is more in the three-dimensional channel than that in the two-dimensional channel. Inorder to study the effects of viscosity ratio, in Fig. 13, we plot the

Fig. 9 Vertical variation of normalized average viscosity, �lxz � lxz=l0xz

� �for (a) h 5 30 deg and

(b) h 5 85 deg. The rest of the parameters are the same as those used in Fig. 6.Here, lxz ¼ 1=LW

RW

0

R L

0 ldxdz and l0xz is the value of lxz at y 5 0.

Fig. 10 Axial variation of normalized average viscosity, �ly ð� ly=l0y Þ, for (a) h 5 5 deg and (b)

h 5 85 deg in a two-dimensional channel. The rest of the parameters are the same as those usedto generate Fig. 6. Here, ly ¼ 1=H

RH

0 ldy and l0y is the value of ly at x 5 0.

Fig. 11 Vertical variation of normalized average viscosity, �lx ð� lx=l0x Þ, for (a) h 5 5 deg and (b)

h 5 85 deg in a two-dimensional channel. The rest of the parameters are the same as those usedto generate Fig. 6. Here, lx ¼ 1=L

R L

0 ldx and l0x is the value of lx at y 5 0.

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spatiotemporal evolution of / contours in the x-y plane for m¼ 5,with the rest of the parameter values the same as those used togenerate Fig. 12. It can be seen that, as the viscosity ratio isdecreased, the interfacial instabilities reduced significantly. Weobserved that the instabilities decrease monotonically withdecreasing the viscosity ratio for this set of parameters. It can beseen in Fig. 14 that increasing Atwood number decreases the ve-locity of the fingertip and makes the nose of the finger “blunt”.The latter effect is also associated with the decrease in Froude

number as compared to Figs. 12 and 13. We found (not shownhere) that the effects of Atwood number, Froude number, andsurface-tension parameter are similar to that of two-dimensionalchannel, and the readers are referred to our previous paper [1] forthe detailed parametric study.

5 Summary

Three-dimensional simulations of pressure-driven flow of twoimmiscible liquids are performed in a square duct using a multi-phase lattice Boltzmann approach and compared with the corre-sponding two-dimensional simulations. A two-phase latticeBoltzmann method (LBM) previously reported by He and co-workers [32] is used to simulate the flow using a three-dimensional, 15-velocity lattice model (D3Q15). We demonstratethat the flow dynamics in a three-dimensional channel is quite dif-ferent than that in a two-dimensional channel. In particular,screw-type instabilities are found in a three-dimensional channel,whereas saw-tooth–type instabilities are apparent in a 2D channel.We also show that increasing angle of inclination increases the in-tensity of the instabilities. We observe that the effects of Atwoodnumber, Froude number, and surface tension are qualitatively sim-ilar to those in a two-dimensional channel. We conclude that dis-placement in an inclined three-dimensional channel is a richproblem, which needs further experimental and computationalinvestigations.

Acknowledgment

K.C.S. thanks the Department of Science and Technology,India for their financial support (Grant No: SR/FTP/ETA-85/2010).

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Fig. 12 Spatiotemporal evolution of the contours of the index function, /. (a) x-y plane atz 5 W/2 (three-dimensional simulation) and (b) the corresponding two-dimensional results. Theparameters are Re 5 100, m 5 20, At 5 0.2, Fr 5 5, h 5 30 deg, and Ca 5 29.9.

Fig. 13 Spatiotemporal evolution of the contours of the indexfunction, /, obtained from the three-dimensional channel form 5 5 with the rest of the parameter values the same as thoseused to generate Fig. 12. The results shown are in x-y plane atz 5 W/2 of the channel.

Fig. 14 The contours of the index function, /, at t 5 20 in thex-y plane at z 5 W/2 for different values of Atwood number.The rest of the parameter values are Re 5 100, m 5 10, Fr 5 1,h 5 30 deg, and Ca 5 29.9.

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