 Zero-liquid-flux surfactant film

In the same gravity–Marangoni reduction, zero liquid flux gives $\Gamma =1+(H_0^2-H^2)/3$ and $6JX=(1+H_0^2/3)(H^3-H_0^3)/3-(H^5-H_0^5)/15$. The lower part of the liquid flows backwards while the surface and upper third flow forwards. For zero initial height and concentration drop $\delta$, $J={\delta }^{3/2}(1-3\delta /5)/(2\sqrt{3})$ is largest formally at complete depletion. Since surface velocity is $J/\Gamma$, that endpoint is singular and requires physical regularization by diffusion, capillarity or an edge model. The algebraic maximum is not by itself a regular zero-concentration outflow.