Hydrothermal Surface-Wave Instability and the Kuramoto-Sivas(8)

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

De ninganew eldby

ζ=c 5

1

4B+3Rζxxxx=0.(24)

Byasimplerescalingofvariables,thisequationcanbewrittenas

ζt+ζζx+ζxx+ζxxxx=0,(25)

whichistheKuramoto-Sivashinsky(KS)equationinitsstandardform[4,5].

ThelinearanalysisperformedbySmithandDavis[1]canberecoveredfromEq.(24)ifweneglectthenonlinearterm,andtake

ζ～exp[ (αx ωt)].

Thisleadstothedispersionrelation

ω= α2R σ

10 α4S

S,weobtainthecriticalReynoldsnumber

Rσ 1

c= 3 10 1

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