dShe

ni2

tuatan- afluBomi

PHYSICAL REVIEW E 67, 036302 ~2003!The Navier-Stokes equation also has its basis in the Boltz-mann equationthe former can be derived from the latterthrough the Chapman-Enskog analysis @14#. That very sameChapman-Enskog analysis can be used to show that the lat-tice Boltzmann methodology can be applied to solve anyconservation law of the continuous Boltzmann equation in-cluding the Navier-Stokes equations. It has also been provedthat the lattice Boltzmann equation tantamounts to an ex-plicit finite difference scheme of the Navier-Stokes equationswith second-order spatial accuracy and first-order temporal

follow a general approach within the framework of kinetictheory for developing the lattice Boltzmann models for mul-tifluid mixtures. This work is a part of our ongoing effort toset the lattice Boltzmann equation on a more rigorous theo-retical foundation and extend its use to more complex flows.We derive a discretized version of the continuum Boltzmannequations for binary mixtures. The extension of this method-ology to multifluid mixtures is relatively straightforward.

Kinetic theory of gas mixtures has received much atten-tion in literature @14,3951#. Many of the kinetic models forgas mixtures are based on the linearized Boltzmann equation@38,52,53#, especially the single-relaxation-time model dueto Bhatnagar, Gross, and Krookthe celebrated BGK model@38#. The kinetic-theory mixtures model employed in thiswork was proposed by Sirrovich @43#, which is also linear innature.

This paper is a detailed follow-up to our previous work

*Electronic address: luo@NIAnet.org; http://research-NIAnet.org/;luo

Electronic address: girimaji@aero.tamu.eduPresent address: National Institute of Aerospace, 144 Research

Drive, Hampton, VA 23666.Theory of the lattice Boltzmann metho

Li-Shi Luo1,*, and1ICASE, Mail Stop 132C, NASA Langley Research Center, 3 W

2Department of Aerospace Engineering, Texas A & M U~Received 11 September 200

A two-fluid lattice Boltzmann model for binary mixkinetic theory by discretizing two-fluid Boltzmann equtreated independently. In the resulting lattice Boltzmvaried independently by a suitable choice of mutualproposed model can simulate miscible and immiscibleThis is in contrast to most existing single-fluid latticescale and hence are restricted to unity Prandtl and Schmultiscalar mixing is quite straightforward.

DOI: 10.1103/PhysRevE.67.036302

I. INTRODUCTION

In many practical flows involving pollutant dispersion,chemical processing, and combustor mixing and reaction,mass and momentum transport within multispecies fluidsplays an important role. For these flows, it can be difficult toconstruct continuum-based models from first principles. Fur-ther, these flows typically involve complex geometry and/ormultiple phases making computation with continuum-basedmodels quite complicated. Therefore, for these flows, there isa growing interest in using the lattice Boltzmann equation@17#.

In general, for computing fluid flow of any type, the lat-tice Boltzmann equation ~LBE! @17# offers several compu-tational advantages over continuum-based methods while re-taining the flow physics intact. Although the origins of themodern LBE can be traced back to lattice-gas automata@810#, the new LBE models are free of some well-knowndefects associated with their predecessors. Recent workshave unequivocally established that the lattice Boltzmannequation is in fact connected to kinetic theory @46# andcompletely consistent with the fundamental conservationprinciples governing fluid flow @1113#. In these papers, apriori derivation of the lattice Boltzmann equation from theparent continuous Boltzmann equation is developed @46#.1063-651X/2003/67~3!/036302~11!/$20.00 67 0363: Two-fluid model for binary mixtures

arath S. Girimaji2,st Reid Street, Building 1152, Hampton, Virginia 23681-2199versity, 3141 TAMU, College Station, Texas 77843-3141; published 21 March 2003!

res is developed. The model is derived formally fromions in which mutual collisions and self-collisions aren model, viscosity and diffusion coefficients can bend self-collision relaxation-time scales. Further, theids by changing the sign of the mutual-collision term.ltzmann models that employ a single-relaxation-timedt numbers. The extension of binary mixing model to

PACS number~s!: 47.11.1j, 47.10.1g, 05.20.Dd

accuracy with a nonzero compressibility @1113#. Thepresent day lattice Boltzmann equation, with its high-fidelityphysics and computation-efficient formulation, is a viable al-ternative to the continuum methods for simulating fluidflows. In fact, it can be argued that for many complex prob-lems involving multifluid phenomena, the physics can bemore naturally captured by the Boltzmann-equation basedmethods rather than Navier-Stokes equation based methods.Recently LBE method has been extended to multiphaseflows @1518# and multicomponent flows @1931#, flowsthrough porous media @32,33#, and particulate suspensions influids @3437#. Most existing LBE models for multicompo-nent fluids @1927# tend to be somewhat heuristic and makethe single-fluid assumption. The single-relaxation-time orBhatnagar-Gross-Krook ~BGK! approximation @38# is usedin most existing models @2127# restricting applicability tounity Prandtl and Schmidt numbers.

A rigorous mathematical development of multifluid latticeBoltzmann equation for multicomponent fluids is still in itsinfancy and such is the object of the present work. As a firststep, in this work we develop a two-fluid lattice Boltzmannmodel which is based on kinetic theory for binary mixtures.Such a model would be capable of ~i! simulating arbitrarySchmidt and Prandtl numbers, and ~ii! accurately modelingthe interaction between miscible and immiscible fluids. We2003 The American Physical Society02-1

L.-S. LUO AND S. S. GIRIMAJI PHYSICAL REVIEW E 67, 036302 ~2003!published as a Rapid Communication in Physical Review E@31# and it is organized as follows. Section II provides a briefreview for some of the existing kinetic models of mixturesthat form the theoretical basis of the present work. SectionIII contains the derivation of the lattice Boltzmann model forbinary mixtures from the corresponding continuous Boltz-mann equations. In Sec. IV, the hydrodynamic equations ofthe lattice Boltzmann model are determined. Section V con-tains the derivation of the diffusion force and the mutualdiffusion coefficient in the lattice Boltzmann model, and thediffusion-advection equations for the mass and molar con-centrations of the system. Section VI discusses the short andlong time behaviors of the model. Section VII concludes thepaper with a summary of the present work and possible di-rections of future work. The three appendixes contain thedetails of: ~a! the iterative procedure to solve the Boltzmannequation; ~b! the discretized equilibrium distribution func-tion; and, ~c! the Chapman-Enskog analysis of the latticeBoltzmann model for binary mixtures.

II. KINETIC THEORY OF GAS MIXTURES

Following a procedure similar to the derivation of theBoltzmann equation for a pure system of single species, onecan derive N simultaneous equations for a system of N spe-cies by reducing the appropriate Liouville equation. For thesake of simplicity without loss of generality, we shall onlydiscuss the Boltzmann equations for a binary system here.The simultaneous Boltzmann equations for a binary systemare

] t f A1j f A1aAj f A5QAA1QAB, ~1a!] t f B1j f B1aBj f B5QBA1QBB, ~1b!

where QAB5QBA is the collision term due to the interactionamong two different species A and B. Obviously, for anN-component system, there will be N such equations, eachcontaining N-collision terms on the right-hand side. In gen-eral, the collision term is

QAB5E djBdV sABijB2jAi@ f 8Af 8B2 f A f B# , ~2!where sAB is the differential cross section of A-B collision,and V is the solid angle, and f 8A( f 8B) and f A ( f B) denote thepost-collision and precollision states of the particle A (B),respectively. Obviously, the equations for a system of mul-tiple species are much more formidable to analyze than thecomparable Boltzmann equation for a pure system of singlespecies. The first modeling objective is to find a suitableapproximation for the collision term given by Eq. ~2! thatwould substantially simplify computation without compro-mising the essential physics.

The justified approximation of the collision terms mustrely on our clear understanding of the underlying physicalsystem. In a system of multiple species, there are a numberof competing equilibration processes occurring simulta-neously. The approach to equilibrium in the system can beroughly divided into two stages. First, each individual spe-03630cies equilibrates within itself so that its local distributionfunction approaches a local Maxwellian distribution. Thisprocess of individual equilibration is also referred to as Max-wellization. Second, the entire system equilibrates so that thevelocity and temperature differences among different speciesvanishes eventually. There are many different time scales inthe equilibrating process of a multicomponent system. In ad-dition, the Maxwellization itself can take place in many sce-narios depending on the molecular weights and mass frac-tions of the participating species. Consider two mixtures,each consisting of a light and heavy gas. In mixture 1, thetotal mass of each species is the same, implying smallernumber density for the heavier gas. In mixture 2, the numberdensities of the two species is the same, implying the massdensity ~or mass fraction! of the heavier species is larger. Inmixture 1, the Maxwellization of light species is mostly dueto self-collision, whereas the equilibration of the heavier spe-cies is predominantly due to cross collisions. This is due tothe fact that the number of molecules of heavy species avail-able for collisions is small. In mixture 2, where the numberdensities of the two species are comparable, Maxwellizationof both species involves self-collision and cross collision.When the process of Maxwellization is complete, the stressof the corresponding species becomes isotropic, or equiva-lently the heat conduction relaxes. Therefore, the time scaleon which the stress becomes isotropic or the heat conductionrelaxes is a suitable measure of Maxwellization.

The equilibration among different species can also takeplace in several different manners. Velocity and temperaturedifferences may equilibrate on the same temporal scale ~as inmixture 1 above! or on vastly different scales ~as in mixture2!. In addition, these equilibrating processes need not to oc-cur sequentially but also concurrently with the Maxwelliza-tion.

There is a significant amount of literature on gas mixtureswithin the framework of kinetic theory @14,3951#. In theChapman-Enskog analysis for a simple gas, one assumes aclear separation of scales in space and time, that is, to dis-tinguish the spatial and temporal scales, which are muchlarger than the mean free path and mean free time. An anal-ogy for a mixture becomes difficult because of multiplicityof length scales. In the classic work of Chapman and Cowl-ing @14#, the full Boltzmann equations ~with integral colli-sion terms! for a binary mixture are analyzed under the as-sumptions that all scales are roughly of the same order, orequivalently, that the phenomenon to be examined is smoothwith respect to all collisional scales. Determination of thevarious transport coefficientsviscosities, diffusivities, ther-mal diffusivities and conductivitywas the main objectiveof that work. However, no attempt was made to describe thedynamics of the evolution.

Direct analysis or computation of the Boltzmann equationis not generally feasible. This is due to the difficulty involvedin evaluating the complex integral collision operators. Tomake further progress one can follow one of two approaches.The first, Grads moment method, is to obtain the non-normal solutions of the Boltzmann equation ~i.e., the solu-tions beyond the hydrodynamic or conserved variables! @54#.Closure modeling would then be required to express the un-closed moments in terms of the closed moments. And the2-2

THEORY OF THE LATTICE BOLTZMANN METHOD: . . . PHYSICAL REVIEW E 67, 036302 ~2003!second is to derive simplified model equations from the theBoltzmann equation, which are more manageable to solve.Many model equations are influenced by Maxwells ap-proach to solving the Boltzmann equation by making exten-sive use of the properties of the Maxwell molecule @55# andthe linearized Boltzmann equation. The simplest model equa-tions for a binary mixture is that by Gross and Krook @41#,which is an extension of the single-relaxation-time model fora pure systemthe celebrated BGK model @38#.

With the BGK approximation @38,41#, the collision inte-grals Qs @s ,P(A ,B)# can be approximated by followinglinearized collision terms:

Jss521ls

@ f s2 f s(0)# , ~3a!

Js521

ls@ f s2 f s(0)# , ~3b!

where f s(0) and f s(0) are Maxwellians

f s(0)5 ns~2pRsTs!D/2

e2(j2us)2/(2RsTs), ~4a!

f s(0)5 ns~2pRsTs!D/2

e2(j2us)2/(2RsTs), ~4b!

where D is the spatial dimension, Rs5kB /ms is the gasconstant of the s species, kB is the Boltzmann constant andms is the molecular mass of the s species. There are threeadjustable relaxation parameters in the collision terms: ls ,l , and ls5(n /ns)ls . The first Maxwellian f s(0) ischaracterized by the conserved variables of each individualspecies: the number density ns , the mass velocity us , andthe temperature Ts ; while the second Maxwellian f s(0) andf s(0) is characterized by four adjustable parameters: us ,us , Ts , and Ts . There are several considerations in de-termining these arbitrary parameters: simplicity of the result-ing theory, accuracy of approximation, and ease of computa-tion. Cross-collisional terms will be symmetric only if onetakes us5us5u and Ts5Ts5T , where u and T are thevelocity and temperature of the mixture. This is essential inpreserving a similarity to irreversible thermodynamics, espe-cially the Onsager relation @56#. On the other hand, fewerterms in the expansion of f s about f s(0) would be needed inmany cases if one chooses us5us and Ts5Ts , i.e.,f s(0)5 f s(0). One salient difference between using u and Tof the mixture in the Maxwellian f s(0) as opposed to usingus and Ts for the species is that the former choice leads to asingle-fluid theory, i.e., a set of hydrodynamic equations forthe mixture, while the latter leads to a two-fluid theory@43,49#, i.e., two sets of hydrodynamic equations for the spe-cies. Obviously, in the cases where the properties betweenthe two species are vastly different, the two-fluid theory ispreferred @56#.

The cross-collision term Js can be better approximatedby expanding f s around the Maxwellian @43#03630Js52f s(0)

nskBTsFmDcs~us2u!1mT 32 S cs

2

2RsTs21 D

3~Ts2T!2M sS cs22RsTs 21 D ~us2u!2G , ~5!where cs5(j2us) is the peculiar ~or thermal! velocity ofthe s species, and

M s5mmrsr

r F 1rs 1mm8 ~ns2n!nsn~ms1m!G , ~6!and mD ,mT ,mm ,mm8 are positive and at most functions ofdensity and temperature @43#, the physical significances ofthese parameters are to be discussed next.

We now consider the following model equations for abinary mixture due to Sirovich @43#:

] t f s1j f s1asj f s5Jss1Js, ~7!where the self-collision term Jss is approximated with theBGK model of Eq. ~3a!, and the cross-collision term Js isgiven by Eq. ~5!. When the external force is not present(as50), and if mD ,mT ,mm ,mm8 are considered as constants~for the sake of simplicity! we can immediately obtain thefollowing moment equations from the above equations:

] t~us2u!52mDS 1rs 1 1rD ~us2u!, ~8a!] t~Ts2T!52mT

1kB

S 1ns

11n

D ~Ts2T!1

23kB

S M sns

2M sn

D ~us2u!2. ~8b!The above equations describe the exponential decay of thevelocity and temperature differences for the two species, asdiscussed in the earlier remarks regarding the processes ofMaxwellianization and equilibration in the mixture. Thephysical significances of the parameters mD , mT , mm , andmm8 become apparent in the above equationsthese param-eters determine the relaxation rates in the Maxwellizationprocesses.

Solving Eqs. ~7! by means of iteration ~cf. Ref. @43# orAppendix A!, one first obtains

us5u5u, ~9a!

Ts5T5T , ~9b!

f s(0)5 f s(0)~ns ,u,T !5ns

~2pRsT !D/2expF2 ~j2u!22RsT G .

~9c!

Substituting the above results into the left-hand side of Eqs.~7!, one has the following equation for the second-order so-lution of f s @43#:2-3

L.-S. LUO AND S. S. GIRIMAJI PHYSICAL REVIEW E 67, 036302 ~2003!f s(0)~ns ,u,T !F nns csds1S csics jRsT 2d i j D Si j1S cs22RsT 2 52 D cs ln TG52

1ls

@ f s2 f s(0)#2 fs(0)

nskBTsFmDcs~us2u!

1mTS cs23RsTs 21 D ~Ts2T!2M sS cs23RsTs 21 D ~us2u!2G , ~10!

where cs5(j2us), and the diffusion force ds and the rate-of-shear tensor Si j are given by

ds5S nsn D1 nsnnr ~m2ms! ln p2 rsrrp ~as2a!,52d , ~11a!

Si j512 ~] iu j1] jui2

23 ud i j!, ~11b!

and p5nkBT is the total pressure of the mixture. From theabove solution of f s, one can compute for the relative ve-locity (us2u), the temperature difference (Ts2T), the~traceless! stress tensor pi j , and the heat flux q @43#:

~us2u!52nkBTmD

ds852n2

nsnDsds8 , ~12a!

~Ts2T!5mDmm8

mT

msm~ns2n!

r~ms1m!~us2u!

2, ~12b!

pi j522kBTS nsls 1nlD Si j2 2mDr S rls 1 rsl D3@~us2u! i~us2u! j2

13 ~us2u!

2d i j# , ~12c!

q552 kBT@ns~us2u!1n~u2u!#

252 kBTS rsls 1 rlD kBT , ~12d!

where the diffusion force

ds85S nsn D1 TsT nsnnr ~m2ms! ln p2 rsrrp ~as2a!,~13!

and the binary diffusion coefficient @53# of the model

Ds5nsnkBT

nmD. ~14!03630The self-diffusion coefficient Dss is a special case of theabove formula when n5n5ns . The viscosity n and ther-mal conductivity k can also be read from the above formulasfor pi j and q as the following:

n5kBTS nsls 1nlD , ~15a!k5

52 kB

2 TS rsls 1 rlD . ~15b!The above transport coefficients are determined by the pa-rameters ls and mD , mT , mm , and mm8 : ls determines theviscosity and the thermal conductivity of the s species andthe combination of lss determines that of the mixture; mDdetermines the diffusion coefficients in the model; andmDmm8 /mT determines the diffusion of the temperature differ-ence due to velocity difference.

Two salient features of the model described by Eqs. ~7!should be addressed. First, the cross-collision term Js of Eq.~5! is exact for the Maxwell molecules obeying the inversefifth-power interaction potential. Equation ~7!, therefore, canbe considered to be a model for the Maxwell gas @43#. Oneimmediate consequence of this approximation is that the dif-fusion force of Eq. ~12a! does not contain a thermal diffusionterm, as it should. Second, the BGK approximation of theself-collision term Jss of Eq. ~3a! imposes the limitation of aunity Prandtl number. However, both these limitations of themodel can be overcome by using the linearized Boltzmannequation @5759# with multiple relaxation times and a non-linear approximation of the collision terms @43,49,60#.

III. THE LATTICE BOLTZMANN MODEL FOR BINARYMIXTURE

We shall construct a lattice Boltzmann model for binarymixtures based on the model given by Eq. ~7!. In the presentwork we only consider the isothermal case such that Ts5T5Ts5T5const. Consequently, we can also ignore theterms related to thermal effects in Js of Eq. ~5! by settingmT5mm5mm8 50, i.e.,

Js521tD

rr

f s(0)RsT

~j2u!~us2u!, ~16!

where the equilibrium function f s(0) for the s species ischosen to be the Maxwellian equilibrium distribution de-pending on the mass velocity of the s species us as thefollowing:

f s(0)5 rs~2pRsT !D/2

expF2 ~j2us!22RsT G . ~17!Note that from hereafter f s(0) and f s are the single particlemass density distribution functions, as opposed to the singleparticle number density distribution functions. We can derivethe lattice Boltzmann equation by discretizing the modelequation ~7!, following the procedure described in Refs.@4,5#:2-4

THEORY OF THE LATTICE BOLTZMANN METHOD: . . . PHYSICAL REVIEW E 67, 036302 ~2003!f as~xi1ead t ,t1d t!2 f as~xi ,t !5Jass1Jas2Fasd t , ~18!

where the self-collision term Jass

, the cross-collision termJa

s, and the forcing term Fa are given by

Jass52

1ts

@ f as2 f as(0)# , ~19a!

Jas52

1tD

rr

f s(eq)RsT

~ea2u!~us2u!, ~19b!

Fas52wars

eaasRsT

, ~19c!

where rs and r , and us and u are the mass densities andflow velocities for species s and , they are the moments ofthe distribution functions:

rs5(a

f as5(a

f as(0) , ~20a!

rsus5(a

f asea5(a

f as(0)ea , ~20b!

and r and u are respectively, the mass density and the bary-centric velocity of the mixture:

r5rs1r , ~21a!

ru5rsus1ru . ~21b!

The collision terms Jass and Ja

s are constructed in such away to respect the mass and momentum conservation laws.~The derivation of the forcing term Fa

s is given in Refs.@17,18#.!

The equilibrium distribution function f as(0) has the follow-ing form in general ~cf. Appendix B!:

f as(0)5F11 1RsT ~ea2u!~us2u!G f as(eq) , ~22a!f as(eq)5warsF11 ~ea2u!uRsT 1 ~eau!22~RsT !2G , ~22b!

where coefficients $wa% depend on the discrete velocity set$ea%. For the sake of concreteness and simplicity withoutlosing generality, we shall restrict ourselves to a nine-velocity model on a two-dimensional square lattice ~D2Q9model!. In this case,

wa5H 4/9, a501/9, a51241/36, a5528.

~23!03630IV. HYDRODYNAMICS

The left-hand side of Eq. ~18! can be expanded in a Taylorseries in d t ~up to second order in d t) and the equation can berewritten as

d tDa f as112 d t

2Da2 f as5Jass1Jas2Fasd t , ~24!

where Da5] t1ea . Obviously,

(a

Jass5(

aJa

s5(a

Fas50, ~25a!

(a

Jassea50, ~25b!

(a

Jasea52

1tD

rsrr

~us2u!, ~25c!

(a

Fasea52rsas . ~25d!

By means of the Chapman-Enskog analysis ~multiple-scaleexpansion!, we can derive the hydrodynamic equations forthe mixture from Eq. ~24! ~see details in Appendix C!.

The mass conservation laws for each species and the mix-ture can be derived immediately from Eq. ~24!:

] trs1~rsus!5 12 FrsrtDr ~us2u!G , ~26!] tr1~ru!50. ~27!

Note that the mass conservation does not hold for each indi-vidual species at the Navier-Stokes level, although it does atthe Euler level. However, the mass conservation law doesapply to the mixture as a whole. The right-hand side of Eq.~26! reflects the mass flux due to diffusion.

We can also derive the Euler equation for each species:

rs] t0us1rsusus52ps1rsas21

tDd t

rsrr

3~us2u!, ~28!

where ps5nskBT5rsRsT is the partial pressure of the sspecies, and the Navier-Stokes equation:

rs] tus1rsusus52ps1rsns2us1rsas2

1tDd t

rsrr

~us2u!, ~29!

where the viscosity of the s species is

ns5RsT~ts212 !d t . ~30!

Equation ~29! is consistent with the results in Ref. @49#.2-5

L.-S. LUO AND S. S. GIRIMAJI PHYSICAL REVIEW E 67, 036302 ~2003!V. DIFFUSION IN ISOTHERMAL MIXTURES

The difference between the two Navier-Stokes equationsfor individual species (s and ) leads to the following equa-tion:

1tDd t

~us2u!52rp

rsrds2$] tdu1~udu!

12~nsus2nu!%, ~31!

where du5(us2u), u5 12 (us1u), (udu)5udu1duu , and the diffusion force

ds5rsrrp F S 1rs ps2 1r pD2~as2a!G

5S nsn

D1S nsn

2rsr D ln p1 rsrrp ~a2as!

5S nsn

D1 nsnnr

~m2ms! ln p1 rsrrp ~a2as!52d , ~32!

where p5nkBT is the total pressure of the mixture, and thetotal number density n is

n5ns1n5rsms

1rm

. ~33!

The diffusion force includes the effects due to the molarconcentration gradient (ns /n), the total pressure gradientand the particle mass difference (ms2m) ln p, and theexternal force (as2a).

It has already been assumed in the derivation of the two-fluid equations that derivatives are slowly varying on thetime scale of Maxwellization @49#. Therefore, the terms in-side the curly brackets Eq. ~31! can be neglected in the dif-fusion time scale. Thus, to the leading order, we have

~us2u!52tDd trp

rsrds . ~34!

Also, by definition @53#, @Eqs. ~6.57a!#, we have

~us2u!52n2

nsnDsds , ~35!

thus the mutual diffusion coefficient in the mixture is

Ds5rkBT

nmsmtDd t . ~36!

Note that the difference between the above formula and Eq.~14! is due to the difference of a factor rsr /r between Ja

s

of Eq. ~19b! and Js of Eq. ~5!, i.e.,

tDd t5rsr

r

1mD

.03630The mass flux of the s species, by definition, is

js5rs~us2u!52tDd tpds ,where we have used the identity that r(us2u)5r(us2u). The continuity equation ~26! for the s species can bewritten as

Dtrs1rsu1js52 12 d t~pds!, ~37!

where Dt5(] t1u). By assuming the incompressibility ofthe fluid ~i.e., u50), we obtain the following advection-diffusion equation for an isothermal mixture:

] trs1urs5tD*d tpds , tD*[~tD2 12 !. ~38!Similar to the Henon correction for the viscosity @61#, thediffusivity is modified by the second-order discrete effect:

Ds* 5rkBT

nmsmS tD2 12 D d t5 rkBTnmsm tD*d t . ~39!

Obviously, the self-diffusion coefficient in the lattice Boltz-mann model for the s species is

Dss* 5kBTms

tD*d t . ~40!

The mass concentration f and molar concentration w ~di-mensionless order parameters! are defined as

f5~rs2r!

~rs1r!, w5

~ns2n!

~ns1n!, ~41!

and they related to each other

f5~ms2m!1~ms1m!w

~ms1m!1~ms2m!w, ~42a!

w5~m2ms!1~m1ms!f

~m1ms!1~m2ms!f. ~42b!

The diffusion force can be written in terms of f and w:

ds512 Fw1~w2f! ln p1r~12f

2!

2p ~a2as!G .The nonlinear diffusion-advection equations satisfied by fand w can be derived from Eq. ~38!:

] tf1uf5 1r ~Df*f1F!, ~43a!

] tw1uw5 An ~Dw*w1F!, ~43b!

where

Df*5ptD*d tmsm

S rnD 25 kBT

msm

r2

ntD*d t , ~44a!2-6

THEORY OF THE LATTICE BOLTZMANN METHOD: . . . PHYSICAL REVIEW E 67, 036302 ~2003!Dw*5ptD*d t , ~44b!

F5@~w2f!p1 12 r~12f2!~a2as!#tD*d t , ~44c!

A512 F ~12w!ms 1 ~11w!m G . ~44d!

Obviously, when ms5m , w5f , and then Eq. ~43a! andEq. ~43b! become identical.

VI. SHORT AND LONG TIME BEHAVIORS

As discussed in Sec. II, short and long time behaviors of abinary mixture involve different Maxwellization and equili-bration processes. This is reflected in the macroscopic equa-tions derived in preceding sections. In an initial stage ~shorttime!, the diffusion velocity (us2u) is significant, thus thesystem is described 2(D11) equations, i.e., two sets ofmass and momentum conservation laws for each speciesgiven by Eqs. ~26! and ~29!. As the system equilibrates sothat the diffusion velocity (us2u) is diminishing, the phys-ics is then described by (D12) equations: the continuityequation of the mixture, Eq. ~27!, the Navier-Stokes equationfor the barycentric velocity of the mixture u, and thediffusion-advection equation for the mass @Eq. ~43a!# or mo-lar @Eq. ~43b!# concentrations. Only in very late stage ofequilibration, the concentration behaves more or less as apassive scalar.

The Navier-Stokes equation for u is

r] tu1ruu52p1P1ra, ~45!where p5kBT(ns1n), ra5rsas1ra , and

P5(s

rsns@~us!1~us!#

~rsns1rn!@~u!1~u!# . ~46!In the derivation of Eq. ~45!, we ignore two terms: one is(tDDs* n/RsTnsn)(pds)2, and the other is in proportionof u(us2u) due to Jas @cf. Eq. ~C7!#. These terms arenegligible when the mixture is more or less homogeneous.Also, when the diffusion velocity (us2u) vanishes, the ap-proximation in Eq. ~46! becomes exact.

Some remarks to place the present work in perspective arein order. Unlike most existing lattice Boltzmann models forbinary mixtures @2227#, the mutual diffusion and self-diffusion coefficients of the present model are independent ofthe viscosity. ~We note that an existing model proposed byFlekkoy @20# already has this feature.! The diffusion coeffi-cients depend on the relaxation parameter tD and relevantphysical properties of the mixture, such as the molecularmasses of each species ms and m , etc. The present model istherefore capable of incorporating more general physics. Inaddition, the present model can simulate both miscible andimmiscible binary mixtures by changing the sign of (tD21/2), i.e., for positive (tD21/2), the mixture is miscible,and for negative (tD21/2), the mixture is immiscible. Wehave performed numerical simulations to verify that: ~a! the03630diffusion coefficient Ds* depends on tD as given by Eq. ~39!and it is independent of ts and t ; and ~b! when (tD21/2),0, spinodal decomposition or separation ~antidiffu-sion! between different species in the mixture occurs. ~Thedetails of the numerical results will be reported elsewhere.!

VII. DISCUSSION AND CONCLUSIONS

We have constructed a lattice Boltzmann model for binarymixtures with several important features. All the modelingissues are addressed at the continuum level within the frame-work of extended kinetic theory. The lattice Boltzmannmodel is then directly derived from the continuous kineticmodel equations using a formal discretization procedure. Thelattice model thus inherits the sound physics and mathemati-cal rigor incumbent in kinetic theory. This is in contrast toprevious lattice Boltzmann models for mixtures @2227#,which are not directly based on the fundamental physics ofcontinuum kinetic equations. These models rely on fictitiousinteractions @22,23# or heuristic free energies @2427# to pro-duce the requisite mixing. ~Many defects of the free-energymodels @2427# are due to the incorrectly defined equilibria@18#.! These nonphysical effects present a further problemsince they are not easily amenable to mathematical analysis@17,18#. The heuristic elements of the previous lattice Boltz-mann models @2227# have been eliminated, resulting in aphysically justifiable model that is simple to compute. Fur-ther, due to the close connection to kinetic theory, the deri-vation of the hydrodynamic equations associated with thelattice Boltzmann model is significantly simplified and ren-dered mathematically more rigorous. The derivation of thehydrodynamic equations from the previous lattice models aremuch less rigorous @2227#.

The second important feature of the present work is thatthe model is based upon a two-fluid theory of binary mix-tures. The previous models @1927,51#, on the other hand,are derived from a seemingly simpler, but highly restrictive,one-fluid theory. In the single-fluid models with BGK ap-proximation one is constrained to use the ad hoc equilib-rium velocity @22,23,51#

u(eq )5~trsus1tsru!

~trs1tsr!

in the equilibrium distribution function f as(0) in order to sat-isfy the local conservation laws. As a result, the viscous re-laxation process and the diffusion process are inseparable.The analysis of these models therefore becomes unnecessar-ily tedious and cumbersome @22,23#. The models with freeenergies @2427# do not yield correct hydrodynamic equa-tions @17,18#, mostly due to the incorrectly defined equilib-rium distribution functions used in these models @18,62#.Furthermore, single-fluid models cannot be applied to mix-tures of species with vastly different properties. In thepresent two-fluid model, the diffusion behavior is decoupledfrom viscous relaxation. The diffusivity is determined by theparameter tD and the physical properties of the mixture. Themodel is capable of simulating either miscible or immisciblefluids by changing the sign of (tD21/2).2-7

L.-S. LUO AND S. S. GIRIMAJI PHYSICAL REVIEW E 67, 036302 ~2003!The proposed LBE model for binary mixtures simulatesdiffusion by considering a mutual interaction term leading tothe diffusion velocity (us2u), which is directly related tothe diffusion driving force in binary mixtures. The diffusionvelocity (us2u) is of the first order in the density gradientr . This suggests that the proposed model, however, doesnot include any higher-order terms of the density gradient.This in turn implies that the proposed model does not have asurface tension, which is related to the density gradientsquare uru2. To include the effect due surface tension, theterms related to uru2 must be explicitly considered.

To further improve the proposed lattice Boltzmann modelfor binary mixing, efforts are currently underway are: ~a!development of a multiple-relaxation-time model for theself-collision term @5759# that will significantly enhance thenumerical stability of the scheme @58,59#; ~b! considerationof models with surface tension; ~c! inclusion of thermal dif-fusion effects which may be important in combustion appli-cations; and, ~d! development of a model for non-Maxwellian molecules. It should be noted that all theexisting lattice Boltzmann models are only applicable to theMaxwell molecules as a direct consequence of the lineariza-tion of the Boltzmann equation. This limitation can be over-come by either using a different expansion of the distributionfunction, or by including the non-Maxwellian effects in thecollision terms.

ACKNOWLEDGMENTS

L.-S.L. is in debt to Professor D. dHumie`res for his criti-cal comments and insights regarding Eqs. ~26! and ~39!, andthe discussion in Sec. VI, is also grateful to Professor V.Sofonea for sharing Ref. @51# before its publication. Thiswork is partly supported by the United States Air Force Of-fice for Scientific Research under Grant No. F49620-01-1-0142.

APPENDIX A: ITERATIVE SOLUTIONOF THE BOLTZMANN EQUATION

We present a short description of the iterative procedure@43# to solve the Boltzmann equation

Dt f 5Q@ f # , ~A1!where Dt5] t1j and Q@ f # is the collision term. The (n11)th iteration solution f (n11) is obtained from the nth it-eration solution f (n) by solving the equation

Q@ f (n11)#5Dt f (n), ~A2!subject to the following conservation constraints

E djf (n11)F 1jj2G5E djf (0)F 1j

j2G , ~A3!

where f (0)5 f (0)(r (n11),u(n11),T (n11)) is the equilibriumdistribution function that depends on the hydrodynamicalmoments r , u, and T computed from f (n11). Following the03630Chapman-Enskog procedure, the temporal derivatives are re-moved by using the conservation equations

] tr(n11)52~ru!,

] tu(n11)52uu1a2 1

rP(n),

] te(n11)52~eu!1rau2q(n)1P(n):u,

where e is the internal energy and e5(D/2)nkBT for idealgases, and P and q are the stress tenor and the heat flux,respectively. On right-hand side of the above equations, thehydrodynamical moments r , u, and e are computed from the(n11)th iteration solution f (n11) but their superscript (n11) are omitted since they are conserved quantities. On theother hand, the stress tensor P and the heat flux q, which arenot conserved, are denoted with the superscript n as they areobtained from the solution of the previous iteration f (n).

In general, the iterative procedure described above is ex-pected to converge more rapidly than a procedure of succes-sive approximation, such as the Chapman-Enskog procedurefor the Boltzmann equation. The reason is that in the iterativeprocedure the ~nonlinear! integral equation must be solved ateach step as given by Eq. ~A2!, whereas in the Chapman-Enskog procedure the integral equation is only solved at theinitial step and at all approximation of higher order only thelinearized integral equation is solved.

APPENDIX B: THE EQUILIBRIUM DISTRIBUTIONFUNCTION

We consider the equilibrium distribution function of Eq.~17! for s species based on its mass velocity us . The distri-bution function can be written in terms of the mixture veloc-ity u as follows:

expF2 12 ~j2us!2

RsTG

5expF2 12 ~j2u!2

RsTGexpF ~j2u!~u2us!RsT G

3expF2 12 ~u2us!2

RsTG . ~B1!

With the procedure described in Refs. @4,5#, f as(eq) of Eq.~22b! is the second-order Taylor expansion of the first expo-nential in the right-hand side of the above equality. Becausethe velocity difference (u2us) is considered to be a smallquantity @49#, the second exponential in the right-hand sideof the above equality can be approximated by its first-orderTaylor expansion:

exp@bs~j2u!~u2us!#11bs~j2u!~us2u!, ~B2!where bs51/RsT . Since the velocity difference (u2us) issmall, the third exponential in the right-hand side that de-pends on (u2us)2 can be neglected in a first-order approxi-mation. It should be noted that this term affects the tempera-2-8

THEORY OF THE LATTICE BOLTZMANN METHOD: . . . PHYSICAL REVIEW E 67, 036302 ~2003!ture difference @cf. Eqs. ~8b! and ~12b!#, and must beincluded if thermal diffusion is important. For the case ofisothermal mixing considered here the simplifications yieldthe final result of the equilibrium distribution function forf ss(0) given by Eq. ~22a! after the velocity space j is properlydiscretized @4,5#.

APPENDIX C: CHAPMAN-ENSKOG ANALYSIS

In the Chapman-Enskog expansion, we introduce a smallparameter ~which is the Knudsen number!, and

f as5 f as(0)1 f as(1)1 , ~C1a!] t5] t01

2] t11 , ~C1b! . ~C1c!

The left-hand side of Eq. ~18! is expanded in a Taylor seriesin d t up to second-order first, and then is substituted with theexpansions of :

f as~xi1ead t ,t1d t!2 f as~xi ,t !5d tDa f as1 12 2d t2DaDa f as15d tDa

(0) f as(0)12d t~] t1 f as(0)1Da

(0) f as(1)

1 12 d tDa(0)Da

(0) f as(0)!1 ,where Eqs. ~C1! have been substituted, and

Da[] t1ea , and Da(0)[] t01ea .The first few equations of a set of successive equations in theorder of obtained from the lattice Boltzmann equation ~18!are

0: f as(0)5 f as(eq)F11 ~ea2u!~us2u!RsT G , ~C2a!1:d tDa

(0) f as(0)521ts

f as(1)1Jas2d tFa , ~C2b!

2:d t~] t1 f as(0)1Da

(0) f as(1)1 12 d tDa(0)Da(0) f as(0)!521ts

f as(2) .~C2c!

In the Chapman-Enskog analysis, it is assumed that the mu-tual interaction term Ja

s as well as the forcing term Fa are ofthe first order in @17,18,63,64#.

The first few moments of the equilibrium distributionfunction f as(0) can be easily computed:

(a

f as(0)5rs , ~C3a!

(a

f as(0)ea5rsus , ~C3b!03630(a

f as(0)eaiea j5RsTrsd i j1rsuiu j1rsF ~us2u ! iu j1~us2u ! jui2

uiu ju~us2u!RsT

GRsTrsd i j1rsuiu j , ~C3c!

(a

f as(0)eaiea jeak5RsTrsD i jklulF12 u~us2u!RsT G1RsTrsD i jkl~us2u ! l1

RsTrs~d i jusk1d jkusi1dkius j!,

~C3d!

where eai is the ith Cartesian component of the vector eaand

D i jkl5d i jdkl1d ikd j l1d ild jk .

In the second- and third-order moments of f as(0) in Eqs.~C3c! and ~C3d!, respectively, the terms involving productsof the velocity and the velocity difference are neglected.

With the substitution of Eq. ~C2b!, Eq. ~C2c! becomes

d t@] t1 f as(0)1Da

(0) f as(1)1 12 Da(0)~Jas2d tFa!#

51ts

@d tDa(0) f as(1)2 f as(2)# . ~C4!

Therefore, the second-order ~in ) equation for the mass con-servation is

] t1rs512 F 1tD rsrr ~us2u!G52 12 d tpds ,

~C5!

where the spatial variation of the external force is neglectedbecause it is canceled out by properly setting the velocitychange due to the external forcing to du5 12 ad t in the equi-librium distribution function @63,64#.

By construction of the forcing term Fa @given by Eq.~19c!#, we have

(a

Faeaiea j50. ~C6!

Therefore, the forcing term has no contribution to the pres-sure tensor. However, the partial pressure tensor for eachindividual species is affected by the cross-interaction termJa

s because

(a

Jaseaiea j52

1tD

rsrr F ~uidu j1u jdui!

21

RsTuiu juduG , ~C7!2-9

L.-S. LUO AND S. S. GIRIMAJI PHYSICAL REVIEW E 67, 036302 ~2003!@18# L.-S. Luo, Phys. Rev. E 62, 4982 ~2000!.@19# A.K. Gunstensen, D.H. Rothman, S. Zaleski, and G. Zanetti,

Phys. Rev. A 43, 4320 ~1991!.@20# E.G. Flekkoy, Phys. Rev. E 47, 4247 ~1993!.@21# D. Grunau, S. Chen, and K. Eggert, Phys. Fluids A 5, 2557

~1993!.@22# X. Shan and G. Doolen, J. Stat. Phys. 81, 379 ~1995!.@23# X. Shan and G. Doolen, Phys. Rev. E 54, 3614 ~1996!.@24# E. Orlandini, W.R. Osborn, and J.M. Yeomans, Europhys. Lett.

32, 463 ~1995!.@25# W.R. Osborn, E. Orlandini, M.R. Swift, J.M. Yeomans, and

J.R. Banavar, Phys. Rev. Lett. 75, 4031 ~1995!.@26# M.R. Swift, E. Orlandini, W.R. Osborn, and J.M. Yeomans,

Phys. Rev. E 54, 5041 ~1996!.036302@45# B.B. Hamel, Phys. Fluids 8, 418 ~1965!.@46# B.B. Hamel, Phys. Fluids 9, 12 ~1966!.@47# L. Sirovich, Phys. Fluids 9, 2323 ~1966!.@48# S. Ziering and M. Sheinblatt, Phys. Fluids 9, 1674 ~1966!.@49# E. Goldman and L. Sirovich, Phys. Fluids 10, 1928 ~1967!.@50# J.M. Greene, Phys. Fluids 16, 2022 ~1973!.@51# V. Sofonea and R.F. Sekerka, Physica A 299, 494 ~2001!.@52# S. Harris, An Introduction to the Theory of the Boltzmann

Equation ~Holt, Rinehart and Winston, New York, 1971!.@53# J.H. Ferziger and H.G. Kaper, Mathematical Theory of Trans-

port Processes in Gases ~Elsevier, New York, 1972!.@54# H. Grad, in Handhuch der Physik, edited by S. Flugge

~Springer-Verlag, Berlin, 1958!, Vol. XII, pp. 205294.@55# J.C. Maxwell, Philos. Trans. R. Soc. London 157, 26 ~1866!.@15# X. Shan and H. Chen, Phys. Rev. E 47, 1815 ~1993!.@16# X. Shan and H. Chen, Phys. Rev. E 49, 2941 ~1994!.@17# L.-S. Luo, Phys. Rev. Lett. 81, 1618 ~1998!.

@42# E.P. Gross and E.A. Jackson, Phys. Fluids 2, 432 ~1959!.@43# L. Sirovich, Phys. Fluids 5, 908 ~1962!.@44# T.F. Morse, Phys. Fluids 6, 1420 ~1963!.where du[(us2u), and dui[(usi2ui). The effect dueto Ja

s is of higher order in u and us , and thus can be ne-glected in the pressure. Consequently, the partial pressuretensor can be approximated as the following:

Pi js(1)5(a

f as(1)eaiea j

52tsd t(a

Da(0) f as(0)eaiea j

52tsd tF] t0(a f as(0)eaiea j1(a ea f as(0)eaiea jG

@1# G.R. McNamara and G. Zanetti, Phys. Rev. Lett. 61, 2332~1988!.

@2# Lattice Gas Methods for Partial Differential Equations, editedby G. D. Doolen ~Addison-Wesley, New York, 1990!.

@3# R. Benzi, S. Succi, and M. Vergassola, Phys. Rep. 222, 145~1992!.

@4# X. He and L.-S. Luo, Phys. Rev. E 55, R6333 ~1997!.@5# X. He and L.-S. Luo, Phys. Rev. E 56, 6811 ~1997!.@6# X. Shan and X. He, Phys. Rev. Lett. 80, 65 ~1998!.@7# D. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice

Boltzmann Models, Lecture Notes in Mathematics Vol. 1725~Springer-Verlag, Berlin, 2000!.

@8# U. Frisch, B. Hasslacher, and Y. Pomeau, Phys. Rev. Lett. 56,1505 ~1986!.

@9# S. Wolfram, J. Stat. Phys. 45, 471 ~1986!.@10# U. Frisch, D. dHumie`res, B. Hasslacher, P. Lallemand, Y.

Pomeau, and J.-P. Rivet, Complex Syst. 1, 649 ~1987!.@11# M. Junk and A. Klar, SIAM J. Sci. Comput. ~USA! 22, 1

~2000!.@12# M. Junk, Numer. Meth. Part. Diff. Eq. 17, 383 ~2001!.@13# M. Junk, A. Klar, and L. -S. Luo ~unpublished!.@14# S. Chapman and T. G. Cowling, The Mathematical Theory of

Non-Uniform Gases, 3rd ed. ~Cambridge University Press,Cambridge, 1970!.52tsd t@] t0~RsTrsd i j1rsuiu j!1RsT~] jrsusi

1] irsus j1d i jrsus!#152tsd t@RsT~] t0rs1rsus!d i j1] t0~rsuiu j!

1RsT~] jrsusi1] irsus j!#12tsd tRsT~] jrsusi1] irsus j!, ~C8!

where the terms smaller than O(M 2) (M is the mach num-ber! are dropped as usual @65#. Therefore,

] jPi js(1)2tsd tRsTrs2usi . ~C9!

@27# A. Lamura, G. Gonnella, and J.M. Yeomans, Europhys. Lett.45, 314 ~1999!.

@28# V.M. Kendon, J.-C. Desplat, P. Bladon, and M.E. Cates, Phys.Rev. Lett. 83, 576 ~1999!.

@29# V.M. Kendon, M.E. Cates, I. Pagonabarraga, J.-C. Desplat, andP. Bladon, J. Fluid Mech. 440, 147 ~2001!.

@30# H. Yu, L.-S. Luo, and S.S. Girimaji, Int. J. Computat. Eng. Sci.3, 73 ~2002!.

@31# L.-S. Luo and S.S. Girimaji, Phys. Rev. E 66, 035301~R!~2002!.

@32# N.S. Martys and H. Chen, Phys. Rev. E 53, 743 ~1996!.@33# S. Chen and G. Doolen, Annu. Rev. Fluid Mech. 30, 329

~1998!.@34# A.J.C. Ladd, J. Fluid Mech. 271, 285 ~1994!; ibid. 271, 311

~1994!.@35# D. Qi, J. Fluid Mech. 385, 41 ~1999!.@36# D. Qi and L.-S. Luo, Phys. Fluids 14, 4440 ~2002!.@37# D. Qi and L.-S. Luo, J. Fluid Mech. ~to be published!.@38# P.L. Bhatnagar, E. P Gross, and M. Krook, Phys. Rev. 94, 511

~1954!.@39# C.F. Curtiss and J.O. Hirschfelder, J. Chem. Phys. 17, 550

~1949!.@40# I. Kolodner, Ph.D. thesis, New York University, New York,

1950.@41# E.P. Gross and M. Krook, Phys. Rev. 102, 593 ~1956!.-10

@56# H. Grad, in Rarefied Gas Dynamics, edited by F. Devienne~Pergamon, London, 1960!, pp. 100138.

@57# D. dHumie`res, Prog. Astronaut. Aeronaut. 159, 450 ~1992!.@58# P. Lallemand and L.-S. Luo, Phys. Rev. E 61, 6546 ~2000!.@59# D. dHumie`res, I. Ginzburg, M. Krafczyk, P. Lallemand, and

L.-S. Luo, Philos. Trans. R. Soc. London, Ser. A 360, 437~2002!.

@60# J.L. Lebowitz, H.L. Frisch, and E. Helfand, Phys. Fluids 3, 325~1960!.

@61# M. Henon, Complex Syst. 1, 763 ~1987!.@62# X. He and G.D. Doolen, J. Stat. Phys. 107, 309 ~2002!.@63# I. Ginzbourg and P.M. Adler, J. Phys. II 4, 191 ~1994!.@64# Z. Guo, C. Zheng, and B. Shi, Phys. Rev. E 65, 046308 ~2002!.@65# X. He and L.-S. Luo, J. Stat. Phys. 88, 927 ~1997!.

THEORY OF THE LATTICE BOLTZMANN METHOD: . . . PHYSICAL REVIEW E 67, 036302 ~2003!036302-11