Simulation of fluid slip at 3D hydrophobic microchannel wall(8)

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

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

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)181–195189

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