Simulation of fluid slip at 3D hydrophobic microchannel wall(6)

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

186L.Zhuetal./JournalofComputationalPhysics202(2005)181–195

Vðx;yÞ¼XX

rr0Grr0ðx;yÞwrðxÞwrðyÞ:0ð10Þ

HeretheGreensfunction,Grr0(x,y),characterizesthenatureoftheinteractionbetweendi erentcompo-nents(attractiveorrepulsiveanditsstrength).Thechoiceofwdeterminestheequationofstateofthesys-temunderstudy.Byselectingdi erentGandw,various uidmixturesandmultiphase owscanbesimulated.Ifonlythenearestneighborinteractionsareconsidered,theGreensfunctionGcanbeputintothefollowingform:

&0;jxÀyj>c;Grr0ðx;yÞ¼grr0;jxÀyj¼c;

wherec=dx/dtisthelatticespeed.Heredxisthespatialwidththendirection,anddtisthetimestep.p along Inourcase,c=0forj=0,c=1forj=1,2,3,4,5,6andc¼2fortheotherdirections.grr0isasymmetricmatrixthatspeci estheinteractionofdi erentcomponentsalongeachdirection.rð0ÞTheequilibriumdistributionfjcanbewrittenas Á3Àrð0Þrrrrrrfjðx;tÞ¼qðx;tÞwj1þ3njÁuðx;tÞþ3njnj:uðx;tÞuðx;tÞÀuðx;tÞÁuðx;tÞ;ð11Þ2

wherewjistheweight,asinthesinglecomponentcase.Themassdensityofcomponentrisde nedbyXrqðx;tÞ¼mrfjrðx;tÞ;ð12Þ

j

wheremristhemolecularmassofcomponentr.Thevelocity,ur,iscomputedvia

dpr

ðx;tÞþsrhrðxÞ;qðx;tÞuðx;tÞ¼qðx;tÞ uðx;tÞþsdtrrrrð13Þ

wheretheaveragevelocityu isde nedby !Xqrðx;tÞXmrXr ¼fðx;tÞnj:uðx;tÞjrrssrrjð14Þ

Heredpr/dtisthenetrateofmomentumchangethatcanbecomputedintermsoftheinteractionpotential

XXXjr0dprrðx;tÞ¼ÀryVðx;yÞ¼ÀwðxÞGrr0wðxþnjÞnj:ð15Þdt0yjr

NotethatheretheGreensfunctiondependsonthedirectionnj.ThisisbecausetheoriginalS-Cmodelwasformulatedona4Dface-centeredhyper-cube(FCHC)lattice.Whenprojectingthe4DFCHClatticeontotheD3Q19lattice,thenearestneighborsinthe4DFCHClatticecorrespondtothenearestandnextnearestneighborsintheD3Q19lattice.

Theforcesh(x)thatweintroducetomodelthehydrophobicwallsareaddedtotheright-handsideoftheequationswhichareusedtocomputetheur.Ourchoiceofhr(x)isasfollows:(index0denotesthe uidinthemodeltosimulatethewaterandindex1the uidtosimulatetheair/watervapor)

h1ðxÞ¼0;

h0ðxÞ¼ð0;g2ðyÞ;g3ðzÞÞ;

g2ðyÞ¼Æg20expðÀy=kÞ;

g3ðzÞ¼Æg30expðÀz=kÞ;