Hydrothermal Surface-Wave Instability and the Kuramoto-Sivas(7)

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

u0= 3

4

θ0=Rσ z2η0ξ;1(17b)z3

2B

S

2.(18)

Atthenextorder,therelevantsolutionsare

u1=RS

160 z 5z54 η0ξξ+6Rzη0ξξξξ 3z2

4η1ξ+1

4η0η0ξ+RS

4η1ξ+1

4 σ2 η0ξξ+2Rη0ξξξξ.(21)

SubstitutingintoEq.(20),weobtainanevolutionequationforη0:

η0τ 5

4B+1S

S= 2Sandη0= 1η,

Eq.(22)reads

ηt+c 5

4B+

13Rηxxxx=0.(23)