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
wherethesubscriptsdenotepartialdi erentiation,uandware,respectively,thexandzcomponentsofthevelocity,andpisthepressure.Ontheupperfreesurface,theseequationsaresubjecttothefollowingboundaryconditions
ηt+uηx=w,
p
(6)ηxx,(7)
(8)2µN32µ1 (ηx)[uz+wx]+2µηx(wz+ux)=N[Tx+ηxTz],
ηxθx+θz=h
2.
Equation(6)isthekinematicboundarycondition,Eqs.(7)and(8)representthenormalandtangentialstressbalanceattheupperfreesurface,andEq.(9)istheequalityoftheheat uxattheinterface,whereθ∞isthetemperatureatz→∞.Onthelowerplate,theboundaryconditionsare
u=w=θz=0,
whichcharacterizesthenoslipandzeroheat uxconditions.
Perturbationtheorywillbeperformedonabasicstateofthesystem,de nedbyimposingtothein niteliquidlayeraconstanthorizontaltemperaturegradient
dθb(10)