Hydrothermal Surface-Wave Instability and the Kuramoto-Sivas(3)

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrotherma

beenreportedintheliteraturetheobservationofsuchwavesinacylindricalcontainer[9].Moreover,hot-wireexperimentsperformedrecently[10,11]haveindicatedthepresenceofpropagativepatterns.Inthe nalsection,wewillspeculateonapossibleconnectionbetweenourresultsandtheseexperiments.

II.MATHEMATICALFORMULATION

Weconsidera uidlayerofheightd,boundedbelow,atz=0,byarigidperfectlyinsulatingplate,andabove,aty=d+η(x,t),byafreedeformablesurfaceincontactwithapassivegasofnegligibledensityandviscosity.Theliquidischaracterizedbyadensityρ,thermalconductivityk,thermaldi usivityκ,unitthermalsurfaceconductanceh,anddynamicviscosityµ.TothefreesurfaceweassociateasurfacetensionT,whichwillbeassumedtodependonthelocaltemperatureθaccordingtothelinearlaw

T=T0 γ(θ θ0),(1)

whereγisapositiveconstant,andT0,θ0arereferencevaluesforsurfacetensionandtem-perature,respectively.

Wewillbeconcernedwithe ectscomingfromthermocapillarityonly.Thereforewewillneglectgravity.Thisisagoodapproximationforathinenoughlayer,oralayerinamicrogravityenvironment.Theequationsgoverningthe uidmotionarewrittenas:

ux+wz=0,(2)

ρ(ut+uux+wuz)= px+µ(uxx+uzz),(3)

ρ(wt+uwx+wwz)= pz+µ(wxx+wzz),(4)

θt+uθx+wθz=κ(θxx+θzz),(5)

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