= Gravity smoothing of a Marangoni front
{title2=$\Delta\sim\rho gh_0^2x_N^2/(AM)$}
When the smoothing width exceeds the <capillary length>, gravity dominates capillarity. Balancing hydrostatic flux $\rho gh_0^4/(\mu\Delta)$ against the <Marangoni stress> flux gives $\Delta\sim\rho gh_0^2x_N^2/(AM)\propto t^{2/3}$. The condition is $\Delta^2\gg\gamma_0/(\rho g)$ in physical units. The smoothing region catches up with the spreading pool at $x_N\sim AM/(\rho gh_0^2)$ and $t\sim\mu(AM)^2/[(\rho g)^3h_0^7]$, after which the sharp-front outer <similarity solution> ceases to apply.
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