Finite-mass Marangoni spreading on a liquid film ($\ell \sim (AM{h}_{0}t/\mu {)}^{1/3}$)

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.

When the smoothing width exceeds the capillary length, gravity dominates capillarity. Balancing hydrostatic flux $\rho g{h}_0^4/(\mu \Delta )$ against the Marangoni stress flux gives $\Delta \sim \rho g{h}_0^2{x}_N^2/(AM)\propto t^{2/3}$. The condition is ${\Delta }^2\gg {\gamma }_0/(\rho g)$ in physical units. The smoothing region catches up with the spreading pool at $x_N\sim AM/(\rho g{h}_0^2)$ and $t\sim \mu (AM{)}^2/[(\rho g{)}^3{h}_0^7]$, after which the sharp-front outer similarity solution ceases to apply.

After gravity smoothing of a Marangoni front reaches the whole spreading pool, the film approaches uniform depth while surface transport continues. A small height gradient creates a hydrostatic pressure return flow that almost cancels the net Marangoni stress flux. Setting $q\simeq 0$ in the thin-film equations with insoluble surfactant gives $h_x\simeq -3A{C}_x/(2\rho g{h}_0)$ and fractional depth variation $O(AM/(\rho g{h}_0^2{x}_N))\ll 1$. The depressed region under the surfactant therefore becomes progressively shallower.

The height jump in linear similarity profiles for surfactant spreading is smoothed by surface tension. If $C_x\sim M/x_N^2$ through a region of width $\Delta$, the Marangoni stress flux is $AM{h}_0^2/(\mu x_N^2)$ and the capillary flux is ${\gamma }_0{h}_0^4/(\mu {\Delta }^3)$. Their balance gives $\Delta \sim ({\gamma }_0{h}_0^2{x}_N^2/(AM){)}^{1/3}\propto t^{2/9}$. This estimate assumes gravity is negligible and the smoothing region remains narrow compared with the pool.

For symmetric finite-mass Marangoni spreading on a liquid film, the leading outer profiles are $h=2h_0|x|/x_N$ and $C=(M/x_N)(1-|x|/x_N)$ inside $|x|<x_N$, where $x_N=(6AM{h}_{0}t/\mu {)}^{1/3}$. Outside, $h=h_0$ and $C=0$. The similarity solution follows from $H{\Gamma }^{\prime }=-\eta /3$, $\eta H^{\prime }=H$, $2\int \Gamma d\eta =1$ and $\int (H-1)d\eta =0$. The film reaches height $2h_0$ immediately behind the front, leaving an outer height jump that gravity or surface tension must smooth. These are ideal leading outer profiles, not a description of the microscopic central dry region or the smoothed edge.