Simulation of fluid slip at 3D hydrophobic microchannel walls by the lattice Boltzmann meth(4)

Fluid slip along hydrophobic microchannel walls has been observed experimentally by Tretheway and Meinhart [Phys. Fluids, 14 (3) (2002) L9]. In this paper, we show how fluid slip can be modeled by the lattice Boltzmann method and investigate a proposed mec

184L.Zhuetal./JournalofComputationalPhysics202(2005)181–195

ThelatticeBoltzmannmethodisanumericaltechniquetosolveasimpli edBoltzmannequation–theLBGKmodel[28,29]

ofðx;n;tÞofðx;n;tÞ1þnÁ¼Àðfðx;n;tÞÀf0ðx;n;tÞÞ;ð1Þwheresistherelaxationtimeandf0istheequilibriumdistributionfunction.ThetermÀ(1/s)(fÀf0)isthewell-knownBGKapproximation[30]tothecomplexcollisionoperatorintheBoltzmannequation.Theparticlevelocityspacencanbediscretizedbya nitesetofvelocities,{nj,j=0,1,2,...n}(inourcase,n=19).Letfj(x,t)bethedistributionfunctionfornj.Thenwehave

ofjðx;tÞofjðx;tÞ1þnjÁ¼Àðfjðx;tÞÀfj0ðx;tÞÞ:otoxs

Afterdiscretizationintime,thelatticeBoltzmannequation(LBE)isobtained

1fjðxþnj;tþ1Þ¼fjðx;tÞÀðfjðx;tÞÀfj0ðx;tÞÞ;sð3Þð2Þ

wherethetermÀð1=sÞðfjÀfj0Þrepresentscollision(notethatcollisionisimplicitlyde nedinLBM,incon-trastwithmoleculardynamicsordirectsimulationMonteCarlo).Beginningwiththeinitialequilibriumdistributionandthedistributionattimet=0,whichcanbetakenastheinitialequilibriumdistribution,theone-stepcomputation(fromtimettotimet+1)canbedividedintotwosubsteps:(1)computethecol-lisionandupdatethedistributionattimetbysummingthecollisiontermandthepre-collisiondistribution;

(2)computethedistributionattimet+1bystreamingthepost-collisiondistribution,i.e.thecomputedrighthandsideoftheLBE.ThelatticeBoltzmannequationcanbetreatedasasecondorderdiscretizationbothintimeandspacebythe nitedi erencemethodoftheLBGKequation.Anyhighorderdiscretizationwilllosetheclearphysicalinterpretationmentionedabove.

AnintuitivewaytoseetheconnectionbetweenthelatticeBoltzmannequationandtheLBGKmodelisasfollows.Followingthetheoryofcharacteristicsforhyperbolicpartialdi erentialequations,letn=dx/dt.TheLBGKequationbecomes

dfðx;tÞ1¼Àðfðx;tÞÀf0ðx;tÞÞ:ð4ÞNotethat(4)isanordinarydi erentialequationalongtheparticletrajectoryinspace(x,t),i.e.(4)anODEinaLagrangiancoordinate.Theprojectionofthetrajectoryonspacexisn=dx/dt.Afterreplacingthetotalderivativein(4)bya nitedi erence(forwardEulermethod),notingthatthediscretizationisdoneinaLagrangiancoordinatesystemandthatdtcanbeabsorbedbys,theLBE(3)isrecovered.Forarig-orousderivationoftheLBEfromthelatticeBoltzmannBGKmodel,see[35,36,38].

Withthenewdistributionfunctionsobtained,themacroscopicquantitiesdensityq(x,t)andmomentumqu(x,t)canbecalculatedateachnodebyXqðx;tÞ¼fjðx;tÞ;ð5Þ

j

ðquÞðx;tÞ¼X

jnjfjðx;tÞ:ð6Þ

Weuseastandard3DlatticeD3Q19whichhas19discreteparticlevelocitiesandcanbewrittenasfollows:8j¼0;><ð0;0;0Þ;

j¼1;2;...;6;nj¼ðÆ1;0;0Þ;ð0;Æ1;0Þ;ð0;0;Æ1Þ;>:ðÆ1;Æ1;0Þ;ðÆ1;0;Æ1Þ;ð0;Æ1;Æ1Þ;j¼7;8;...;18: