= Balanced scales for a thermocapillary film
{title2=$\epsilon^2=3Bh_*/(2\gamma_0),\quad \widehat t=4\mu\gamma_0/(3B^2h_*)$}
Write the surface tension as $\gamma_s=\gamma_0+Bh$, where $B>0$. The <thermocapillary thin-film equation> has flux $\gamma_0h^3h_{xxx}/(3\mu)+Bh^2h_x/(2\mu)$. Balancing both terms at thickness scale $h_*$ gives aspect ratio $\epsilon^2=3Bh_*/(2\gamma_0)$ and time scale $4\mu\gamma_0/(3B^2h_*)$. The fractional surface-tension variation is $Bh_*/\gamma_0=2\epsilon^2/3$, so using constant tension in the leading <capillary pressure> term is consistent with the balanced long-wave limit.
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