= Axisymmetric viscous gravity current
{title2=$h_t=r^{-1}(rh^3h_r)_r$}
A shallow radially spreading viscous drop on a horizontal plane has lubrication flux proportional to $-h^3h_r$. Absorbing the positive dimensional coefficient into time gives the displayed equation. For a finite-volume drop, $M=\int_0^Rrh\,dr$ is constant when the origin and front carry zero volume flux. A similarity form $h=t^{-\beta}f(r/t^\alpha)$ has $2\alpha=\beta$ from volume and $\beta+1=4\beta+2\alpha$ from the equation, giving $\alpha=1/8$ and $\beta=1/4$.
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