= Linear similarity profiles for surfactant spreading
{title2=$x_N=(6AMh_0t/\mu)^{1/3}$}
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 $x_N=(6AMh_0t/\mu)^{1/3}$. 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.
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