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)