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

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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

JournalofComputationalPhysics202(2005)

181–195

/locate/jcp

Simulationof uidslipat3Dhydrophobicmicrochannel

wallsbythelatticeBoltzmannmethodq

LuodingZhu

ba,*,DerekTrethewayb,LindaPetzolda,b,CarlMeinhartbDepartmentofComputerScience,UniversityofCaliforniaSantaBarbara,SantaBarbara,CA93106,USADepartmentofMechanicalandEnvironmentalEngineering,UniversityofCaliforniaSantaBarbara,SantaBarbara,CA93106,USA

Received15September2003;receivedinrevisedform15April2004;accepted5July2004

Availableonline14August2004

Abstract

FluidslipalonghydrophobicmicrochannelwallshasbeenobservedexperimentallybyTrethewayandMeinhart

[Phys.Fluids,14(3)(2002)L9].Inthispaper,weshowhow uidslipcanbemodeledbythelatticeBoltzmannmethodandinvestigateaproposedmechanismfortheapparent uidslip[Phys.Fluids(2003)].Byapplyinganexponentiallydecayinghydrophobicrepulsiveforceof4·10À3dyn/cm3withadecaylengthof6.5nm,ane ective uidslipof9%ofthemainstreamvelocityisobtained.Theresultisconsistentwithexperimentall-PIVdataandwiththeproposedmechanism.

Ó2004ElsevierInc.Allrightsreserved.

Keywords:Fluidslip;Slipboundarycondition;Hydrophobicity;Micro uidic;LatticeBoltzmannmethod

1.Introduction

Inclassical uidmechanics,theassumptionofno-slipatasolidboundaryisusedastheboundarycon-ditionforviscous owsatrigidwalls.However,for owsatmicro-andnanoscales,thisassumptionmaynolongerbeaccurate.Manyresearchershaveinvestigatedthe uidslipphenomenon[3–12,14,21].Choietal.

[3]investigatedexperimentallytheslipe ectsofwater owinhydrophilic/hydrophobicmicrochannelsandfoundthesliplengthtovaryapproximatelylinearlywiththe owshearrate.Lummaetal.[4]measuredthe owpro lenearawallbydouble-focus uorescencecross-correlation;theiranalysisyieldsalargeapparent uidslipatthewall.Hornetal.[5]observedthehydrodynamicslippage,whichwasdeducedfromthin lmq

*ThisworkwassupportedbyNSF/ITRACI-0086061.Correspondingauthor.

E-mailaddress:zhuld@cs.ucsb.edu(L.

Zhu).

0021-9991/$-seefrontmatterÓ2004ElsevierInc.Allrightsreserved.doi:10.1016/j.jcp.2004.07.004

182L.Zhuetal./JournalofComputationalPhysics202(2005)181–195

drainagemeasurementsinasolutionofnonadsorbingpolymer.Watanabeetal.[6,7]found uidslipatthewallofahydrophobicduct/pipewithrelativelylargescalegeometry(15·15mm).RuckensteinandRajora

[8]studiedthe uidslipinaglasscapillarywithliquid-repellentsurfaces.Alargeslipwasinferredatthewallfrompressuredropversus owmeasurements.BarratandBocquet[9]predictedcomputationallysig-ni cantslipinnanoporousmedia.Thiswascon rmedexperimentallybyChuraevetal.[10].ZhuandGra-nick[11]studiedexperimentallytheslipinanoscillatingsurfaceforceequipment.Pitetal.[12]investigated uidslipbetweenspinningparalleldisks.ThompsonandTroian[13]simulatedNewtonianliquidsundershear,viamoleculardynamics.Theirresultsuggestedthatthereisanonlinearrelationshipbetweenthemagnitudeofslipandthelocalshearrateatasolidsurface.Foracomprehensivereviewof uidslippageoverhydrophobicsurfaces,see[14]andthereferencestherein.Thehydrophobicityphenomenaarenotwellunderstood.Forreaderswhoareinterestedinhydrophobicity,werefertothefollowingpapersandrefer-encestherein:[15–21].

Recently,TrethewayandMeinhart[1]measuredthevelocitypro lesofdeionizedwater owingthrougha3Dmicrochannelwithacross-sectionof30·300lm.Theyfoundthatwhenthemicrochannelsurfaceishydrophilic(thewallattractswatermolecules),theconventionalassumptionofano-slipbound-aryconditionisvalid.However,whenthemicrochannelsurfaceishydrophobic(thewallrepelswatermol-ecules),asigni cantslip(approximately10%ofthefree-streamvelocity)nearthewallwasmeasured.Thevelocityerrorintheexperimentalmeasurementiswithin2%,andthesliplengtherroriswithin±0.45lm.

Inthispaper,wedescribethenumericalsimulationofthe uidsliponhydrophobicmicrochannelwallsusingthelatticeBoltzmannmethod.Inthe rstpartofourwork,wereportcomputersimulationswiththesinglephase(component)latticeBoltzmannmethod(LBM)for owin3Dmicrochannels,focusingonmodelingoftheslipboundarycondition.Inthesecondpart,weaddressthemechanismof uidslipwiththemultiphase(multicomponent)latticeBoltzmannmethod(theS-Cmodel).Wewanttopointoutthat,inbothcases,weaddressmodelingofthe uidslipgeneratedbyhydrophobicityinwater ow,notthe uidslipgeneratedbyKne ectsforgas ow.

2.Numericalmethods–latticeBoltzmannmethods

ThelatticeBoltzmannmethodisanalternativetotraditionalnumericalmethodsforsolvingincompress-ibleNavier–Stokesequations.Insteadofsolvingforthemacroscopicquantitiesvelocityandpressure(orstreamfunctionandvorticity)directly,LBMdealswiththesingleparticlevelocitydistributionfunctionsf(x,n,t)(xrepresentsthespatialcoordinates,ntheparticlevelocitycomponents,andtisthetimevariable)basedonasimpli edBoltzmannequation.ForapplicationofthelatticeBoltzmannmethodintheareaofmicroscale ows,see[27,40,41].

Inthe rstpartofourwork1wefocusonmodelingtheslipboundaryconditionusinga19-discretevelocitylatticeBoltzmannmodel(D3Q19)[28,29].InthelatticeBoltzmannmethod,thebounce-backschemeisusuallyusedtomodeltheno-slipboundarycondition.(Wenotethatthebounce-backschemecanitselfalsogenerateslip.Ananalysisoftheslipgeneratedbybounce-backforsimple owscanbefoundin[37].Wefoundthat,ona neenoughgrid,theamountofslipcausedbythebounce-backschemealoneisnegligiblecomparedtotheamountobservedinexperiment.Knudsennumberrelatedslipusingthebounce-backschemeformicroscale owcanbefoundin[40,41].)Ithasalsobeensuggestedintheliteraturethatspecularre ectionmaybeusedtomodelaslipboundarycondition.However,thespecularre ectionschemeusedinourworkresultedin100%slipofthe uidonthewalls.Instead,wehaveemployedacom-

Preliminaryresultshavebeenpresentedatthe2002ASMEInternationalMechanicalEngineeringCongress&Exposition,NewOrleans,Louisiana,November,2002.See[25].1

L.Zhuetal./JournalofComputationalPhysics202(2005)181–195183

binationofbounce-backandre ectiontosimulatetheslipboundarycondition.Thus,theslipboundaryconditionismodeledbyassigningaprobabilityofqforbounce-backand1Àqforspecularre ectionwhenaparticlevelocitydistributionfunctionfj(x,t)hitsawall(100%bounce-backifnjisperpendiculartoawall).Byvaryingthevalueofq,di erentslipboundaryconditionsmaybemodeled.Anobviousshortcomingofthisapproachisitslackofpredictabilityoftheamountofslip.AsimilarschemehasbeenusedbySucci[55]tostudyslipmotionat uidsolidinterfaceswithheterogeneouscatalysis.Suchanideawasmentionedin

[34,35],andcanbedatedbackto1867whenMaxwellstudiedthemicroscopicmodelingofthesolidbound-ary[42].Itwasonceusedin[43]totreattheno-slipboundaryconditioninthelatticegasmethod,andhasalsobeenmentionedinthecontextofdirectsimulationMonteCarlo(DSMC)[44].(InDSMC,specularre ectioniscombinedwiththefulldi usioncondition.)Thecombinationofbounce-backandspecularre ectionisdi culttoimplementinacomplexgeometry.Amethodforaddressingthatissuehasbeenpro-posedin[45].

Inthesecondpartofourwork,wemakeuseofthemulticomponentlatticeBoltzmannmethod[49–52]toinvestigateapossiblemechanism[2]forgeneratingtheapparent uidsliponahydrophobicwall.Thegen-eralideaofthemechanismisasfollows.Thewaterusedinthelaboratoryexperimentwasnotdegasedandcontainsasmallamountofabsorbedgas.Thehydrophobicwallmayrepelthewatermoleculeswithinaregionveryclosetothewallbutisneutraltowatervaporandairmolecules.Asaresult,thewaterdensitynearthewallmaydecline,creatingadepletedlayerveryclosetothewall.Thus,athinlayerofwater–air/watervapormixturewithsigni cantlydi erentwaterandair/watervapordensities(comparedtothewell-mixedair–waterunderstandardconditions)mayformintheregionveryneartothewall.Becausetheva-pordensityismuchsmallerthanthatofwater,theaveragedensityofthethinlayerdeclinescomparedtotheaveragemixturedensityelsewhere.Sincethepressuredropbetweentheinletandoutletthatdrivesthe owcanbetreatedasapproximatelyconstantoncross-sectionsoftheinletandoutlet,thethinlayermaymovefasterthantheusuallymixedwater–air(e.g.,inthecaseofahydrophilicwall),whichmayresultinapparentsliponthehydrophobicwall.

Wetestedtheabovepropositionbysimulatingthewater–air/watervaportwo-phasesystemwiththemulticomponentlatticeBoltzmannmethodfor owina3Dhydrophobicmicrochannel.Thehydrophobicwallsweremodeledbyapplyingforcesinaregionveryclosetothewalls.Theseforcesarerepulsivetothewatermolecules,andareneutraltotheair/watervapormoleculedistributionfunctions.Theseforcesexpo-nentiallydecayawayfromthewall.Theinitialwater–airmixtureisassumedtobeuniform.Theinitialden-sityoftheairinthewateriscalculatedunderstandardconditions(at20°Cand1atm).ThemulticomponentlatticeBoltzmannmodelweuseistheS-Cmodel,see[49–52],exceptthatweintroducedtheadditionalhydrophobicwallforcesintotheformulation.Thewallforcetermwasinsertedintotheright-handsideoftheequationswhichareusedtoupdatethevelocitiesforcomputingthenewequilibriumveloc-itydistributions.

Numerousresearchershaveexaminedhydrophobicsurfacesandtherelatedforces.Whilethee ectsofhydrophobicforceshavebeenobserved,theformandmagnitudeofthehydrophobicforceisnotwellunderstood.Asa rstapproximation,wemodeledthehydrophobicforceasasimpleexponentialdecaywithamagnitudeandadecaylength.AsimilarforcefunctionwasexploredbyVinogradova[21].Wesetthemagnitudeanddecaylengthtobeconsistentwithexperimentalobservations.Thedecaylengthisconsistentwiththeexperimentallengthscalesatreducedviscositylayer(5nm)[23,16]ornanobubbles(10–30nm)[22],aswellasthevalueassumedbyVinogradova(decaylength5–10nm)[21].ThemagnitudeisthreeorderssmallerthanthatassumedbyVinogradova[21].

2.1.SinglecomponentlatticeBoltzmannmethod

TheLBMsusedinourworkareinthe rstpartthesinglephaseisothermalLBGKmodel[28,29],andinthesecondpartthemultiphaseS-Cmodel[49,50].

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

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:

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

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

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;

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

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

188L.Zhuetal./JournalofComputationalPhysics202(2005)181–195

withre nementratio2:100·50·5,200·100·10and400·200·20.Theratiowas3.89.Thisindicatesthatthenumericalmethodisofsecondorderinspace,asclaimedintheliterature.Duetocomputationallimitations,wedidnotcheckthison nergrids.Instead,wecomparedthevelocity eldpresentedinourpapertothosecomputedfromaseriesof nergrids(500·250·25,600·300·30,700·350·35),andfoundthattheyarealmostindistinguishable.

ThesimulationwasperformedonthePC-clusteroftheComputerScienceDepartmentatUCSB.Theclusterhas33dual-processor(IntelXeon)nodesthatareconnectedby1GBcopper.Eachnodehasamem-oryof3GBandeachprocessorhasaspeedof2.6GHz.ThelatticeBoltzmannmodelsweusedwerepar-allelizedbydomaindecompositionandMPI.See[24]fordetails.Thesimulationwasrununtilthe owreachedsteadystate(approximately500,000steps).Theconventionalbounce-backschemeinLBMwasap-pliedtomodeltheno-slipboundarycondition,whileacombinationofbounce-backandre ectionwasem-ployedtosimulatetheslipboundarycondition.

Intheno-slipcase,ournumericalsolutionmatchesverywelltheexactsolutionofStokes owassumingano-slipboundarycondition[26],andalsoagreeswellwiththeexperimentalresult.Fig.2showstheveloc-itypro lesinthecaseofhydrophilicwalls.Thepro leistakenatthecross-sectionx=300lmwiththezcoordinateequalto15lm.Thex-axisisthenormalizedvelocity,andthey-axisisthedistancefromthewall(unit:micron).Thesquaresaretheexperimentaldata,thedashedlineistheexactsolution,andthesolidlineistheLBMsimulationresult.Wecanseethatoursimulationresultisalmostindistinguishablefromtheexactsolutionandmatcheswellwiththeexperimentaldata.

Intheslipscenario,ournumericalvelocitypro leagreeswellwiththatoftheexperiment.Aslipofabout10%onthewallwasattainedbyassigningtheprobabilityofbounce-backto0.03andofre ectionto0.97whenthevelocitydistributionfunctionhitsthewall.SeeFig.3forvelocitypro lesinthecaseofhydro-phobicwalls.Thepro leistakenatthecross-sectionx=300lmwiththezcoordinateequalto15lm.Thex-axisisthenormalizedvelocityandthey-axisisthepositionfromthewall.Thetrianglesrepresentexperimentaldata.ThesolidlineistheLBMsimulationresult.Wecanseethatournumericalresultagreesreasonablywellwiththeexperimentaldata.

InFig.4,bothvelocitypro lesalongtheyandzdirectionswereplottedtogether.Thepro lesaretakenatthecross-sectionx=300lmwiththezcoordinateequalto15lmasafunctionofy,andwiththey

L.Zhuetal./JournalofComputationalPhysics202(2005)181–195189

Fig.2.Velocitypro lesinthecaseofhydrophilicwalls.Thepro leistakenatthecross-sectionx=300lmwiththezcoordinateequal15lm.Thex-axisisthenormalizedvelocityandthey-axisisthedistancefromthewall(unit:micron).Thesquaresaretheexperimentaldata,thedashedlineistheexactsolution,andthesolidlineistheLBMsimulationresult.

Fig.3.Velocitypro lesinthecaseofhydrophobicwalls.Thepro leistakenatthecross-sectionx=300lmwiththezcoordinateequalto15lm.Thex-axisisthenormalizedvelocityandthey-axisisthepositionfromthewall.Thetrianglesrepresentexperimentaldata.ThesolidlineistheLBMsimulation

result.

coordinateequalto150lmasafunctionofz.Thex-axisisthenormalizedvelocityandthey-axisisthedistancefromthewalls,normalizedbythedepthandwidthofthechannel,respectively.Thesolidlineisthepro lealongtheydirection,andthecurveplottedbytrianglesisthepro lealongthezdirection.Wecanseethatthe uidslipinthezdirection(channeldepth)isslightlylargerthantheslipintheydirection(chan-nelwidth).Experimentaldataarenotavailableforthevelocitypro lealongthedepthdirection.

Fig.5showsthesliplengthasafunctionoflocationalongthewidthdirectionandthedepthdirection.Fig.5(a)plots uidsliplengthatthetoporbottomwallsasafunctionofdistancefromthesidewallalongthewidthdirection,andFig.5(b)plots uidsliplengthatthesidewallsasafunctionofdistancefromthebottomwallalongthedepthdirection.Weseethatthevariationofsliplengthalongthesidewalls(sepa-ratedby300lm)issigni cantlydi erentfromthevariationofsliplengthalongthebottomandtop

walls

190L.Zhuetal./JournalofComputationalPhysics202(2005)181–195

Position on bottom channel wall (microns)25020015010050

011.21.4

Slip length (microns)Position on side channel wall (microns)Slip length (y-direction)300Slip length (z-direction)30252015105(c)(a)011.21.4(b)Slip length (microns)

Fig.5.(a)and(b)depicthowthe uidsliplengthvariesalongtheperimeterofthechannelatstreamwisepositionofx=300lm.(a)Variationofsliplengthasafunctionofyalongthetoporbottomwall.(b)Variationofsliplengthasafunctionofzalongthesidechannelwalls.(c)Measurementsampleplaneandthelocationsofsliplengthplottedin(a)and(b).

(separatedby30lm).However,themagnitudesaresimilar,rangingbetween1.1and1.4lm.Fig.5(c)showsthemeasurementsampleplaneandthelocationsofsliplengthplottedinFigs.5(a)and(b).

3.2.Multi-componentlatticeBoltzmannsimulation

Inthesecondpartofourwork,weinvestigatedapossiblegeneratingmechanismforapparent uidslip

[2],viathemulti-componentlatticeBoltzmannmethod(theS-Cmodel).Weperformedthesimulationona0.1·1·2lm3microchannel.Thegridspacingis5nm.Thenon-dimensionalhydrophobicwallforceusedinthesimulationis0.2,correspondingtoaphysicalforceof4·10À3dyn/cm3withadecaylengthof6.5

L.Zhuetal./JournalofComputationalPhysics202(2005)181–195191

nm,aswasspeci edinSection2.Theappropriatemagnitudeofthisforceisnotwellde ned.However,Vinogradova[21]modeledattractivehydrophobicinteractionsasadecayingexponentialwithamagnitudeof1dynandadecaylengthofbetween5and15nm.Forthecurrentsimulation,theforcefunctionwaschosensothatthesimulationresultswouldbeconsistentwithexperimentalobservations.Whilethedecaylength,k=6.5nm,isconsistentwiththevaluesofVinogradova[21],themagnitudeofthehydrophobicforce,4·10À3dyn,issigni cantlylower.Thedi erencemayarisefrompossiblenon-uniformitiesinthehydrophobicOTScoatingsinthemicrochannels.Thisrepulsivehydrophobicforcecausesthedensityofthesynthetic uidusedtosimulatewaterinthemulti-componentlatticeBoltzmannsimulationtobegreaterthan1.Werescaledthedensityto1forthe uidusedtomodelwaterbythemaximumdensityinthesim-ulationresult(about1.07).

Weperformedsimulationsonaseriesofgraduallyre nedgrids.Thenumberofnodesinthezdirectionwas10,15,20,25,30.Wefoundthatthe uidslippercentagewasconvergentasthegridwasre ned.

Fig.6showsthe uiddensitiesasafunctionofdistanceawayfromthesidewallatthecross-sectionx=1lmandz=50nm.Thex-axisisthedensityandthey-axisisthedistancefromthesidewall.Fig.6(a)showsthedensityofthe uidusedtosimulatewaterinthemodelalongtheydirection(inthemiddleofthezdirection)onacross-sectioninthemiddleofthechannel(xdirection).Fig.6(b)showsthedensityofthe uidusedtosimulatewatervapor/air.Wecanseethatthedensityofwaterisdecreasedandthatofwatervapor/airisincreasedclosetothewalls.Fig.7givesadetailedpictureofthedensitychangeclosetothewall.Sakuraietal.[20]havealsoobservedadrasticdecreaseofthewatermoleculenumberdensityatamonolayer–waterinterfacefromthesimulationresultsofwaterbetweenhydrophobicsurfaces,viamoleculardynamics.Ourresultsareconsistentwiththeirs.

Fig.8showsthenormalizedstreamwisevelocitypro leandalocalblowupalongtheydirectionatcross-sectionx=1lmforz=50nm.Thex-axisisthenormalizedvelocity,andthey-axisisthepositionfromthesidewall(unit:micron).Thesolidline(in(a)and(b))isthevelocitypro lewhennowallforcesarepresent.

192L.Zhuetal./JournalofComputationalPhysics202(2005)181–195

Thedottedline(inpart(a)),orthedashedline(inpart(b))isthecasewherewallforcesareintroduced.Incontrasttotheformercase,thelatterresultsinapparentslipatthewalls.(SeeFig.8(b)forthelocalblowupnearthesidewall.)WecanseefromFigs.6–8thatintheregionveryclosetothewalls,thewaterdensitydecreasesandthewatervapor/airdensityrises.Thisenablesthe uidsliponthewalls(approximately9%offreestreamvelocity)comparedtothesolidlinesinFig.8,whichillustratethecasewherenohydrophobicwallforceswereapplied.

L.Zhuetal./JournalofComputationalPhysics202(2005)181–195193

4.Summaryanddiscussion

WiththesinglephaselatticeBoltzmannmethod(D3Q19model),wesimulatedthe owofwaterina3Dmicrochannelwithhydrophilic/hydrophobicwalls.Theclassicbounce-backschemewasusedtomodelthehydrophilicwalls,whileacombinationofbounce-backandspecularre ectionwasappliedtomodelthepartialslipboundaryconditionatthehydrophobicwalls.Goodquantitativeagreementwasobservedbe-tweenthesimulationsandpreviousexperimentalresults.Inthecaseofhydrophilicwalls,thesimulationresultagreesalmostexactlywiththeanalyticsolution.Inthecaseofhydrophobicwalls,a10%slipwasattainedbyassigningtheprobabilityofbounce-backto0.03andtheprobabilityofre ectionto0.97.ThevalueofqisconsistentwithSucciÕswork[55].Thisseemstoindicatethatpartial uidslipgeneratedbyhydrophobicitymaybemodeledbyacombinationofbounce-backandspecularre ection.However,itremainstobefurtherveri edwhetherthecombinationschemecanaccuratelycapturetheslipmotioncausedbyhydrophobicity.See[55]fordetails.

WiththemultiphaselatticeBoltzmannmethod(theS-Cmodel),weinvestigatedapossiblemechanismfor uidslip.Duetocomputationallimitations,thecorrespondingphysicalsizeofthesimulationdomainwas0.1·1·2lm,whereastheexperimentalresultswereobtainedinamicrochannelwithacross-sectionof30·300lm.Thehydrophobicwallsweremodeledbyapplyinganexponentiallydecayingforceof4·10À3dynwithadecaylength6.5nmfromthewall,whichisconsistentwiththeworkofVinogradona

[21].Thisforcerepelsthewatermolecules,buthasnoe ectontheair/watervapormolecules.Theforceproducesaslipofapproximately9%ofthemainstreamvelocity,whichcorrespondstotheexperimentall-PIVresults.Itindicatesthatthepresenceofadepletedwaterlayer(lowdensityregion)nearthehydro-phobicsurfacemayproducetheapparent uidslipobservedexperimentally.

Acknowledgment

WethankthefollowingpeoplefortheirusefulconversationsanddiscussionsofthelatticeBoltzmannmethod:ShulinHou,ShiyiChen,HudongChen,NicosMartys,XiaoboNie,andLishiLuo.

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