= Capillary smoothing of a Marangoni front
{title2=$\Delta\sim(\gamma_0h_0^2x_N^2/(AM))^{1/3}$}
The height jump in <linear similarity profiles for surfactant spreading> is smoothed by <surface tension>. If $C_x\sim M/x_N^2$ through a region of width $\Delta$, the <Marangoni stress> flux is $AMh_0^2/(\mu x_N^2)$ and the capillary flux is $\gamma_0h_0^4/(\mu\Delta^3)$. Their balance gives $\Delta\sim(\gamma_0h_0^2x_N^2/(AM))^{1/3}\propto t^{2/9}$. This estimate assumes gravity is negligible and the smoothing region remains narrow compared with the pool.
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