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