Simulation of fluid slip at 3D hydrophobic microchannel wall(5)

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

L.Zhuetal./JournalofComputationalPhysics202(2005)181–195185

Forisothermal uids,theequilibriumdistributionfunctionfj0(whichisafunctionofqandu)intheD3Q19latticecanbecomputedvia 932ð7Þfj0ðx;tÞ¼qðx;tÞwj1þ3njÁuðx;tÞþðnjÁuðx;tÞÞÀuÁu;22

wherewjistheweight,whichtakesthevalues:

8><1=3;j¼0;

wj¼1=18;j¼1;2;...;6;>:1=36;j¼7;8;...;18:

Weusetheconventionalbounce-backschemetomodeltheno-slipboundarycondition.Weuseacombi-nationofbounce-backandspecularrefectiontomodeltheslipboundarycondition;thatis,whenaparticlevelocitydistributionfunctionfjhitsawall,fjisbouncedbackwithprobabilityq,andisre ectedwithprob-ability1Àq.Anyfjwhichhitsawallalongitsnormaldirectionisbouncedback.There ecteddistributionfunction,fj,goestowardsaneighboringnodewhichis±dxawayfromtheoriginalnode,andupdatesthedistributionfunctionattheneighboringnodealongthedirectionitisre ected.Inoursimulation,boththebounce-backandre ectionareexecutedwhenadistributionfunctionishalfwaybetweenitsoriginalsiteandawall.Otherwise,theorderofaccuracymaysu erneartheboundaries.Exceptformodelingoftheslipboundarycondition,thesinglecomponentLBMusedinoursimulationisthelatticeBKGD3Q19model.ReadersinterestedinLBmethodscansee[31–33,35,36,39,56]andthereferencestherein.Wewanttopointoutthattheconceptsofbounce-backandspecularre ectionmaynothavedirectphysicalanalogsforliquids.Theyareusedhereasanidealizationandsimpli cationofthephysics.Theprobabilitiesofbounce-backandre ectionarearami cationofthecomputation.Theyarenotbaseddirectlyonexper-imentalorphysicalresults.

2.2.Multi-componentlatticeboltzmannmethod

TherecurrentlyexistseveralversionsofthemulticomponentlatticeBoltzmannmethod:theR-Kmodel

[46,47],theS-Cmodel[48–51],Swift[57],He[58],Seta[59],Inamuro[60],Luo[61].TheS-CmodelhasbeentestedinthestaticcasebyHouetal.[53]andNiimura[63].Ithasbeensuccessfullyappliedtosimulatedropletdeformationundershear owina3Dchannel,seeXiandDuncan[62].Hereinourworkitisusedtosimulatemultiphase owwithtwocomponents.Foreach uidcomponentr(r=0,1inourcase),asin-gleparticlevelocitydistributionfunctionfr(x,n,t)isintroduced,whichsolvestheLBGKmodelforthatcomponent

ofrðx;n;tÞofrðx;n;tÞ1þnÁ¼Àrðfrðx;n;tÞÀfrð0Þðx;n;tÞÞ:otoxsð8ÞHeresrandfr(0)aretherelaxationtimeandtheequilibriumdistributionfunctionforcomponentr,respectively.

Afterdiscretizationinparticlevelocityspacenandintimet,wegetthemulticomponentLBE

1rrð0Þðfðx;tÞÀfðx;tÞÞ;ð9Þjjsr

wherefjristhedistributionfunctionforthercomponentalongthedirectionnj.Notethatthediscretizationinnisthesameforeachcomponent.

OneimportantnewfeatureforthemulticomponentlatticeBoltzmannmodelistheintroductionofaninterparticleinteractionpotential,whichisde nedasfjrðxþnj;tþ1Þ¼fjrðx;tÞÀ