= Finite-mass Marangoni spreading on a liquid film
{title2=$\ell\sim(AMh_0t/\mu)^{1/3}$}
A localized amount $M=\int C\,dx$ of <insoluble surfactant> spreads over a planar film of depth scale $h_0$. Neglect gravity, capillarity and diffusion. Then $C\sim M/\ell$ and <Marangoni stress> drives $u_s\sim AMh_0/(\mu\ell^2)$, giving $\ell\sim(AMh_0t/\mu)^{1/3}$. The <thin-film equations with insoluble surfactant> admit a <similarity solution> in which the film height and concentration depend on $x/\ell$, conserving total <surfactant> and fluid volume relative to the undisturbed layer.
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