Molecular Dynamics Simulation of Compressible Flui

arXiv:cond-mat/9205013v126May1992February1,2008Moleculardynamicssimulationofcompressiblefluidflowintwo-dimensionalchannelsM.SunandC.EbnerDepartmentofPhysicsTheOhioStateUniversityColumbus,OH43210AbstractWestudycompressiblefluidflowinnarrowtwo-dimensionalchannelsusinganovelmoleculardynamicssimulationmethod.Inthesimulationarea,anupstreamsourceismaintainedatconstantdensityandtemperaturewhileadownstreamreservoiriskeptatvacuum.Thechannelissufficientlylonginthedirectionoftheflowthatthefinitelengthhaslittleeffectonthepropertiesofthefluidinthecentralregion.Thesimulatedsystemisrepresentedbyanefficientdatastructure,whoseinternalelementsarecreatedandmanipulateddynamicallyinalayeredfashion.Consequentlythecodeishighlyefficientandmanifestscompletelylinearperformanceinsimulationsoflargesystems.Weobtainthesteady-statevelocity,temperature,anddensitydistributionsinthesystem.Thevelocitydistributionacrossthechannelisverynearlyaquadraticfunctionofthedistancefromthecenterofthechannelandrevealsvelocityslipatthe1boundaries;thetemperaturedistributionisonlyapproximatelyaquarticfunctionofthisdistancefromthecentertothechannel.Thedensitydistributionacrossthechannelisnon-uniform.Weattributethisnon-uniformitytotherelativelyhighMachnumber,approximately0.5,inthefluidflow.Anequationforthedensitydistributionbasedonsimplecompressibilityargumentsisproposed;itspredictionsagreewellwiththesimulationresults.Validityoftheconceptoflocaldynamictemperatureandthevariationofthetemperaturealongthechannelarediscussed.PACSnumbers:47.40.Dcand47.60.+i2I.IntroductionThetechniqueofmoleculardynamics(MD)simulationhasbeenwidelyusedtostudynon-equilibriumfluids.Becauseoflimitationsimposedbyfinitecomputationalcapacitywithregardtobothmemoryandspeed,thismethodhasbeenusedprinci-pallytodeterminethebehaviorsoffluidsystemsontimeanddistancescaleswithinafewordersofmagnitudeofτcandrc,respectively.Here,τc≈10−13secisthecollisiondurationandrc≈10−8cmisamolecularsize.1Onlymolecularpropertiesofthefluidcanbeobtainedinthisrangeoftimeanddistance,andthatisalsothedomainofexperimentalneutronscatteringmeasurements.Realisticexaminationofthehydrodynamicpropertiesofflowisstillbeyondthereachofmostmoleculardy-namicssimulations,althoughtherearemanyattemptstosimulatelargersystemsonlongertimescales.AmongthesearetheworkofKoplik,etal.,2,3Hannon,etal.,4andBhattacharyaandLie.5,6Inparticular,Hannon,etal.,haveobtainedvelocityandtemperaturedistributionsacrosschannelsinwhichflowisoccurring.Theirre-sultsagreewellwithsimplehydrodynamicpredictionsforincompressiblefluidflow.Also,BhattacharyaandLiehaveobtainedsimilarvelocityprofilesandhavecomputedboundaryslipcoefficients.Thesesimulationsoffluidflowhaveacommonproperty.Periodicboundarycon-ditionsareintroducedalongtheflowdirectionintheinterestofreducingtheamountofcomputationneededtoobtainusefulresults.Theimpositionoftranslationalin-variancealongtheflowdirectionmakesitnecessarytointroducea“gravitational”fieldtoinduceflow.Inordertoinduceappreciableflow,thisfieldmustbegivenastrengthmuchlargerthantheearth’sfieldg,forexample,aslargeas21012g.Asaconsequenceofthegravitationalfield,regularrescalingoftheparticles’kineticen-3ergiesisrequired;henceonecannotreliablystudypropertiesofthefluidhavingtodowithenergyorheatflow.Also,itisnotpossibletostudyvariationsoftheflowpropertiesalongtheflowdirection.Inthispaper,wepresentanewmethodofsimulationinwhichthechannelisoffinitelengthintheflowdirectionwithoutperiodicboundaryconditions.Atoneendofthechannelisasourceregionwhichismaintainedatconstantdensityandtemperaturebyintroducingparticlesasneeded.Attheotherendisasinkregionwhichismaintainedatvacuum;thatis,anyparticlewhichmovesintothisregionisremovedfromthesystem.Hencethepressureordensitygradientalongthechannelisprimarilyresponsibleforinstigationoftheflow.Wealsouseanovellayereddatastructuretoimprovetheefficiencyofthecodeandtheutilizationofstorage.Ourmethodsmakeitpossibletosimulatesystemscontaining20,000ormoreparticlesformorethan106timestepsonaDECstation3100.SectionIIcontainsadescriptionofthesystemsimulatedandofournumericaltechniques.TheresultsareinSec.III,andSec.IVcontainsadiscussionandconclu-sions.II.ModelandnumericalmethodsThegeometryofthesystemisindicatedinFig.1.ThechannelhasalengthLinthexdirectionandwidthwintheydirection.Flowisalongthexdirectionwiththesourceregionat0xL1andthevacuumorsinklocatedatxL.Thechannelisclosedatx=0sothatparticlescannotescapeintotheregionx0.TypicalchannelsizesthatwehaveinvestigatedareL=400σ,w=100σ,andL1=100σ,whereσis4theparticlesizeparameter(diameter)intheLennard-JonespotentialV(r)=4ǫσr12−σr6#.(1)whichwehaveemployedfortheinterparticleinteractions.Theinteractionistrun-catedatr≥2σ.7Inadditiontoǫandσ,theonlyotherparameterdescribingthepropertiesofthemoleculesisthemassm.Wehaveusedvaluesappropriateforar-gon,i.e.,σ=3.4rA,ǫ/k=119.76K,andm=6.67×10−23g;kistheBoltzmannconstant.Wealsointroduceabasictimeconstantτ=pmσ2/48ǫ≈3×10−13sec.Ourprocedureforhandlingcollisionsoftheparticleswiththewalls,whicharethesurfacesx=0,y=0,andy=w,istogivetherecoilingparticlesaMaxwell-Boltzmannvelocitydistributiononahalf-space.Forexamp