Axisymmetric viscous gravity current ($h_t=r^{-1}(r{h}^3{h}_r{)}_r$)

A shallow radially spreading viscous drop on a horizontal plane has lubrication flux proportional to $-h^3{h}_r$. Absorbing the positive dimensional coefficient into time gives the displayed equation. For a finite-volume drop, $M={\int }_0^{R}rhdr$ 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 $\beta =1/4$.