Hydrothermal Surface-Wave Instability and the Kuramoto-Sivas(4)

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)

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