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Onset of thermal convection in a horizontal layer of granular gas

The Navier-Stokes granular hydrodynamics is employed for determining the threshold of thermal convection in an infinite horizontal layer of granular gas. The dependence of the convection threshold, in terms of the inelasticity of particle collisions, on th

Onsetofthermalconvectioninahorizontallayerofgranulargas

EvgeniyKhainandBaruchMeerson

RacahInstituteofPhysics,HebrewUniversityofJerusalem,Jerusalem91904,IsraelTheNavier-Stokesgranularhydrodynamicsisemployedfordeterminingthethresholdofther-malconvectioninanin nitehorizontallayerofgranulargas.Thedependenceoftheconvectionthreshold,intermsoftheinelasticityofparticlecollisions,ontheFroudeandKnudsennumbersisfound.AsimplenecessaryconditionforconvectionisformulatedintermsoftheSchwarzschild’scriterion,well-knowninthermalconvectionof(compressible)classical uids.Themorphologyofconvectioncellsattheonsetisdetermined.AtlargeFroudenumbers,theFroudenumberdropsoutoftheproblem.AstheFroudenumbergoestozero,theconvectioninstabilityturnsintoarecently2002

cdiscoveredphaseseparationinstability.

棋牌游戏eDPACSnumbers:45.70.Qj

棋牌游戏6 INTRODUCTION

]tfFluidizedgranularmediaexhibitaplethoraoffas-so.cinatingpattern-formationphenomenathathavebeentasubjectsofmuchrecentinterest[1].Inthisworkwemaddressthermal(buoyancy-driven)granularconvection-[2,3,4,5,6].Beingunrelatedtotheshearortime-ddependenceintroducedbythesystemboundaries,itre-nosemblestheRayleigh-B`enardconvectioninclassical uidcanditscompressiblemodi cations10,11,12].In[classical uidconvectionrequiresanexternallyimposed 2negativetemperaturegradient,thatisatemperaturegra-vdientinthedirectionoppositetogravity.Inavibro u-3idizedgranularmediumanegativetemperaturegradient2setsinspontaneouslybecauseoftheenergylossbyin-4elasticcollisions.Convectiondevelopswhentheabsolute80valueofthetemperaturegradientislargeenough.Inthe2simplestmodelofinelastichardspheresthatwewilluse0ithappenswhentheinelasticitycoe cientq=(1 r)/2/texceedsacriticalvaluedependingontherestofthepa-arametersofthesystem.Hereristhecoe cientofnormalmrestitutionofparticlecollisions.

-dThermalgranularconvectionwas rstobservedinnmoleculardynamics(MD)simulationsofasystemofinelasticallycollidingdisksinatwo-dimensional(2D)cosquareboxTheboundariesoftheboxdidnot:vintroduceanyshearortime-dependence:thesystemwasiXdrivenbyastress-freethermalizingbase.Thetopwallwasperfectlyelastic,whilethelateralboundarieswereraeitherelasticorperiodic.Experimentwithahighly u-idizedthree-dimensionalgranular owgivesstrongev-idenceforthermalconvection,thoughenergylossatthesidewallsintroducescomplicationsAcleariden-ti cationofthermalconvectioninexperimentrequiresalargeaspectratiointhehorizontaldirection,sothatmul-tipleconvectioncellscanbeobserved.MDsimulationsin2Dofavibro uidizedgranularsystemwithalargeaspectratioindeedshowmultipleconvectioncells[4].Thisworkdealswithatheoryofthermalgranularconvectioninasystemwithalargeaspectratio.Re-cently,acontinuummodelofthermalgranularconvection

棋牌游戏hasbeenformulatedintheframeworkoftheNavier-Stokesgranularhydrodynamics.InthedilutelimittheNavier-Stokeshydrodynamics(or,moreprecisely,gasdy-namics)issystematicallyderivablefrommorefundamen-talkineticequations[13,14].Likeanyotherhydrody-namicapproach,theNavier-Stokeshydrodynamicsde-mandssmallKnudsennumbersforitsvalidity.Inaddi-tion,ithasbeenshownthatatmoderateinelasticitiesqnon-hydrodynamice ects(suchasthelackofscalesepa-ration,thenormalstressdi erenceandnon-Gaussianityintheparticlevelocitydistribution)maybecomeimpor-tant[15].Therefore,theNavier-Stokesgranularhydro-dynamicsisexpectedtobeaccuratequantitativelyonlyfornearlyelasticcollisions,q 1.Thoughrestrictive,thenearlyelasticlimitisconceptuallyimportant.Also,onecanexpectsomeoftheresults,obtainedinthislimit,tobestillqualitativelyvalidforlargerinelasticities,suchasthoseencounteredinexperiment.

棋牌游戏InRef.[5]thefullsetofnonlinearhydrodynamicequa-tionsforthermalgranularconvectionwassolvednumer-icallyina2Dboxwithaspectratio1.Itwasobserved,inqualitativeagreementwithMDsimulationsthatthestaticstateofthesystem(asteadystatewithazeromean ow)giveswaytoconvectionviaasupercriticalbifurcation,theinelasticityqbeingthecontrolparame-ter.Thepresentworkemploysthesamehydrodynamicformulationtoperformasystematiclinearstabilityanalysisofthestaticstate.Wedeterminetheconvec-tionthresholdasafunctionofthescaledparametersoftheproblemandofthehorizontalwavenumberofsmallperturbations.Thisanalysismakesitpossibletopredicttheconvectionthresholdanddeterminethemorphologyoftheconvectioncellsinasystemwithanarbitraryas-pectratio,includinganin nitehorizontallayer,astan-dardsettingforconvectioninclassical uidsWealsoformulateasimplenecessary(butnotsu cient)cri-terionforthermalgranularconvectionintermsoftheSchwarzschild’scriterion,well-knowninthermalconvec-tionof(compressible)classical uidsFinally,wetakethelimitofazerogravityandestablishtheconnectionbetweenthermalconvectionandarecentlydiscovered

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