Simulation of fluid slip at 3D hydrophobic microchannel wall(7)

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)181–195187

whereyisthedistanceawayfromthesidewallsalongtheinwardnormaldirection,andzhasasimilarmeaningforthetopandbottomwalls.Thekcanbedeterminedbyspecifyingthedistanceawayfromthewall(y0andz0)wheretheforcediminishesto1%ofthemaximummagnitude(g20andg30)atthewall:0.01g20=g20exp(Ày0/k).Inthepresentsimulation,theparameterswerechosenasfollows:g20=g30=0.2(4·10À3dyn/cm3indimensionalform.WeaddressthischoiceinSection3),y0=30nm,z0=30nm.Thedecaylengthisk=6.5nm

wr¼qr; 0Gj

rr0¼0:2 0¼Gj

0rr0:1!0:20!0:10forj¼0;1;...;6;forj¼7;8;...;18:

ThesechoicesofwandGhavebeenusedintheliterature,inthesimulationofbubblesin uids[49,52,53].Thevaluesofy0andz0werechosentobeconsistentwiththebubbleheightsobservedexperimentallybyTyrellandAttard[22].

Themacroscopicquantitiesareconnectedtodistributionfunctionsbythefollowingrelations:Xqðx;tÞ¼qrðx;tÞ;ð16Þ

r

ðquÞðx;tÞ¼X

rmrXjfjrnj1Xdprðx;tÞ:þ2rdtð17Þ

Thedimensionlessviscosityofthesystemisde nedbyP2qrsr

rÀ1:m¼6ð18Þ

3.Simulationresults

3.1.SinglecomponentlatticeBoltzmannsimulation

Inthe rstpartofourwork,wecloselyfollowedtheparametersintheexperiment[1],exceptthatthelengthofthechannelwasdecreasedfrom8.25cmintheexperimentto600lminthesimulation.Inthesimulation,thelengthofthechannel(600lm)wastwicethewidthofthechannel(300lm).Theshorterchannellengthisjusti edbecauseaperiodicboundaryconditionisusedalongthechanneldirection.SeeFig.1foradiagramofthe3Dmicrochannelusedinthesimulation.Themicrochannellengthdirectionisdenotedasthexdirection(600lminthesimulation),thewidthastheydirection(300lminthesim-ulation),andthedepthasthezdirection(30lminthesimulation).Inallthe gurespresentedbelow,thevelocitypro leplottedwastakenonthecross-sectionx=300lmataplanewithz=15lmnormaltothecross-sectionasafunctionofy,orataplanewithy=150lmnormaltothecross-sectionasafunc-tionofz,dependingonthecontext.Thesimulationpresentedhereusesaspatialdiscretizationwithreso-lution400·200·20(x,y,zdirections,respectively).

Weperformedsimulationsonaseriesofgraduallyre nedgrids.Thenumberofnodesinthezdirectionwas10,15,20,25,30,35.ThelatticeBoltzmannsimulationsondi erentgridswereperformedaccordingtothepaper[54].Wefoundthatthe uidslippercentagewasconvergentasthegridwasre ned.Wealsocom-putedthequantityiu2hÀu4hiL2/iuhÀu2hiL2(wherehisthegridspacing)onthreesuccessivelyre nedgrids