= Zero-liquid-flux surfactant film
{title2=$Q=0$}
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 $\delta$, $J=\delta^{3/2}(1-3\delta/5)/(2\sqrt3)$ 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.
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