= Thermocapillary thin-film equation
{title2=$h_t+\partial_x(a_ch^3h_{xxx}+a_mh^2h_x)=0$}
= Thermocapillary film instability
{synonym}
For a liquid film heated below and cooled above, the <quasistatic temperature of a conductively cooled film> makes a thicker region colder at the surface. If <surface tension> decreases with temperature, <Marangoni stress> draws fluid towards thicker regions, competing with smoothing by <capillary pressure>. Writing $\gamma_s=\gamma_0+bh$ gives $a_c=\gamma_0/(3\mu)$, $a_m=b/(2\mu)$. A balanced long-wave scaling gives $H_\tau+(H^2H_X)_X+(H^3H_{XXX})_X=0$. The <dispersion relation> $s=k^2-k^4$ has unstable band $0<|k|<1$ and fastest <normal mode> at $|k|=1/\sqrt2$. This thermal mechanism does not require <surfactant>.
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