 Finite-mass Marangoni spreading on a liquid film

A localized amount $M=\int Cdx$ 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 $u_s\sim AM{h}_0/(\mu {\ell }^2)$, giving $\ell \sim (AM{h}_{0}t/\mu {)}^{1/3}$. 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.