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1、PhysicaD135(2000)175–194Aphase-?eldmodelofsolidi?cationwithconvectionD.M.Andersona,G.B.McFaddenb;,A.A.WheelercaDepartmentofMathematics,UniversityofNorthCarolina,ChapelHill,NC27599,USAbMathematicalandComputationalSciencesDivision,NationalInstituteofStandardsandTechnology,Gaithe
2、rsburg,MD20899-8910,USAcFacultyofMathematicalStudies,UniversityofSouthampton,High?eld,SouthamptonSO171BJ,UKReceived29October1998;accepted15March1999CommunicatedbyH.Müller-KrumbhaarAbstractWedevelopaphase-?eldmodelforthesolidi?cationofapurematerialthatincludesconvectionintheliqu
3、idphase.Themodelpermitstheinterfacetohaveananisotropicsurfaceenergy,andallowsaquasi-incompressiblethermodynamicdescriptioninwhichthedensitiesinthesolidandliquidphasesmayeachbeuniform.Thesolidphaseismodeledasanextremelyviscousliquid,andtheformalismofirreversiblethermodynamicsise
4、mployedtoderivethegoverningequations.Weinvestigatethebehaviorofourmodelintwoimportantsimplesituationscorrespondingtothesolidi?cationofaplanarinterfaceatconstantvelocity:densitychange?owandashear?ow.Intheformercaseweobtainanon-equilibriumformoftheClausius–Clapeyronequationandinv
5、estigateitsbehaviorbybothadirectnumericalintegrationofthegoverningequations,andanasymptoticanalysiscorrespondingtoasmalldensitydifferencebetweenthetwophases.Inthecaseofaparallelshear?owweareabletoobtainanexactsolutionwhichallowsustoinvestigateitsbehaviorinthesharpinterfacelimit
6、,andforlargevaluesoftheviscosityratio.?2000ElsevierScienceB.V.Allrightsreserved.PACS:05.70.Fh;05.70.Ln;81.10.Mx;47.20.HwKeywords:Phase-?eld;Convection;Solidi?cation;Diffuseinterface1.IntroductionDiffusetheoriesofinterfacesseparatingtwobulkphasesweredevelopedinthe19thcenturybyPo
7、isson[1],Gibbs[2],Maxwell[3],Rayleigh[4]andvanderWaals[5].Previously,interfaceshadbeenmodeledbyYoung,LaplaceandGaussassurfaces.Inthelatterformulationtheinterfaceisregardedasasingularsurfaceonwhichassociatedphysicalmechanismsarelocalizedandrepresentedasboundaryconditionstobeappl
8、iedatthesurface;e.g.,thenotionofsurfaceenergyasanenerg
