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1、2.BasicEquations2.1VolumeEquationsTheflowvelocitiesUinbuoyancy-orsurface-tension-drivenflowsclosetothefirstpattern-forminginstabilityaretypicallysmallcomparedtothespeedofsoundc.Asagoodapproximationthefluidflowcanthenbeconsid-eredincompressiblewithaconstantdensityp.Correctionsareoftheo
2、rderofmagnitudeO(M2),whereM=U/cdenotestheMachnumber(seee.g.Schlichting(1968)).Themomentum,mass,andenergyconservationequa-tionsforanincompressiblefluidwithkinematicviscosity~,thermaldiffusiv-ity~,andthermalexpansioncoefficient~=-1/p(Op/OT)pinahomogeneousgravityfieldg=-gezare(Landau&:Li
3、fshitz(1959))Otu+u.Vu=-1Vp+,Au+gl~Tez,(2.1)PV.u=0,(2.2)OtT+u.VT=~AT,(2.3)where(2.2)isthecontinuityequation.Thevelocity,pressure,andtemper-aturefieldsaredenotedbyu,p,andT,respectively.Equations(2.1-2.3)arecalledtheOberbeck-Boussinesqequations.Withinthisapproximationtemperature-inducedd
4、ensityvariationsareretainedonlyinthebuoyancytermg~Tez.Theapproximationrequiresthetemperaturevariationstobesufficientlysmall.1TheOberbec~Boussinesqequationsarethebasisforthefollowinganalysesofbuoyantandthermocapillaryconvection.2.2BoundaryConditions2.2.1VelocityFieldThefluidvolumesunde
5、rconsiderationareboundedbyrigidstationarywallsaswellasbyfreesurfaces.Sinceafluidincontactwitharigidwallmaynotslip,thevelocitycomponentsparallelandperpendiculartothewallmustvanish:1MoreaccuratecriteriahavebeengivenbyGray&:Giorgini(1976).122.BasicEquationsu=0.(2.4)Alongthefreesurfacebet
6、weentwoimmisciblefluids(1)and(2)theforcesonadjacentsurfaceelementsof(1)and(2)mustbethesame.Ifthesurfaceisplaneandthesurfacetensionisconstant,thisbalancereadsS(1) 9n=S(2) 9n,(2.5)wherethepressureandtheviscousforcesperunitsurfaceareaenterthestresstensorS=S~j=-p~j+~lOxj+Ox~/-pl+UHere,U=
7、P~isthedynamicviscosity,Iistheidentitymatrix,andnistheunitnormalvectordirectedoutofliquid(1)intotheambientfluid(2),whichisassumedtobegaseous.Generally,thefreesurfaceisnotplane,andthesurfacetensionvariesalongtheinterface.Inthiscasetheappropriateboundaryconditioniss(1) 9n+n)n-(,-nn).w=s
8、(2)-(
