= Steady surfactant film with zero surface flux
{title2=$J=0$}
In the steady nondiffusive gravity–Marangoni model with capillarity neglected, zero surfactant flux and positive concentration make the surface <velocity> zero. With $Q=-H^2\Gamma_X/2-H^3H_X/3$ and $J=-H\Gamma\Gamma_X-H^2\Gamma H_X/2$, integration gives $H^4=H_0^4-48QX$ and $\Gamma=1+(H_0^2-H^2)/4$. A concentration drop $\delta$ over unit length therefore selects $Q=-\delta(H_0^2+2\delta)/6$. The liquid flows opposite to the imposed Marangoni traction, with <velocity> proportional to $H_XY(Y-H)/2$ and an immobilized surface. This reduction is singular at a dry endpoint; its small-slope assumption does not resolve that edge.
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