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
III.PERTURBATIONTHEORY
Letusconsiderthefollowingperturbationstothebasicstate:
u=ub+ (u0+ u1+....),(13a)
w= 2(w0+ w1+...),(13b)
p=pb+ (p0+ p1+...),(13c)
θ=θb+ (θ0+ θ1+...),(13d)
η=ηb+ (η0+ η1+...),
where isasmallparameter.Next,weintroducetheslowvariablesaccordingto
ξ= (x ct),(13e)(14)
τ= 2t,(15)
andwesupposethattheperturbationsdependon(x,t)through(ξ,τ)only.Wenowmakemoreprecisethemeaningoflargealbeit niteforS,byassumingthat
S= 2
S～O( 0).
Weareable,atthispoint,toobtainanorderbyordersolutiontotheproblembyintegrationsinthezvariable.Ateachorder,thebulkequationsandtheboundaryconditionsatthebottomyieldu,w,θ,pintermsofthreearbitraryfunctions,whichwillbedeterminedintermsofη0bytheboundaryconditionsattheupperfreesurface.However,aswehavefourequations,acompatibilityconditionwillariseateachorder.Explicitly,atorder 0,weget: