= Gravity-levelled surfactant film
{title2=$h_x\simeq-3AC_x/(2\rho gh_0)$}
After <gravity smoothing of a Marangoni front> reaches the whole spreading pool, the film approaches uniform depth while surface transport continues. A small height gradient creates a <hydrostatic pressure> return flow that almost cancels the net <Marangoni stress> flux. Setting $q\simeq0$ in the <thin-film equations with insoluble surfactant> gives $h_x\simeq-3AC_x/(2\rho gh_0)$ and fractional depth variation $O(AM/(\rho gh_0^2x_N))\ll1$. The depressed region under the <surfactant> therefore becomes progressively shallower.
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