= Thin-film equations with insoluble surfactant
{title2=$h_t+q_x=0,\quad C_t+(Cu_s)_x=0$}
For a no-slip planar liquid film with an <insoluble surfactant>, <lubrication theory> gives $q=-h^3p_x/(3\mu)+h^2\gamma_x/(2\mu)$ and surface <velocity> $u_s=-h^2p_x/(2\mu)+h\gamma_x/\mu$. <Mass conservation> and <conservation of insoluble surfactant on a moving interface> give $h_t+q_x=0$ and $C_t+(Cu_s)_x=0$. With $\gamma=\gamma_0-AC$ and small slopes, $p_x=\rho gh_x-(\gamma h_{xx})_x$. The different coefficients in $q$ and $u_s$ are essential: <Marangoni stress> and hydrostatic return flow transport fluid and <surfactant> differently.
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