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