 Gravity-levelled surfactant film

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 $q\simeq 0$ in the thin-film equations with insoluble surfactant gives $h_x\simeq -3A{C}_x/(2\rho g{h}_0)$ and fractional depth variation $O(AM/(\rho g{h}_0^2{x}_N))\ll 1$. The depressed region under the surfactant therefore becomes progressively shallower.