Hydrothermal Surface-Wave Instability and the Kuramoto-Sivas(5)

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

temperature,γβastheunitofpressure,andT0astheunitofsurfacetension.Inthisprocessthefollowingdimensionlessnumbersappear:

R=ργβd2

ρκ,B=hd

µ2,

whereRistheReynoldsnumber,σisthePrandtlnumber,BistheBiotnumber,andSisthesurfacetensionnumber.Inaddition,wehavealsotheMarangoninumberM,whichisde nedbyM=Rσ.

Thebasicstate,fromwhichwewillconsidersmalldisturbances,isthereturn owsolutiontotheEqs.(2-10)which,inthelimitS→∞,isgivenby:

ub=3

2

pb=

z3 (12a)wb=0;

161 z4 1

4Sx3+R

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