= Viscous gravity current on a cone
Let $x$ be slant distance from an upward apex, and let the surface slope down at angle $\alpha$ to the horizontal. Axisymmetric <lubrication theory> gives
$$
h_t+\frac Kx\partial_x\left[xh^3(\sin\alpha-\cos\alpha\,h_x)\right]=0,
\qquad K=\frac{\rho g}{3\mu}.
$$
For conserved volume $V$ and $x_N^3\gg V/\sin\alpha$, the leading <similarity solution> has $h=\sqrt{x/(5K\sin\alpha\,t)}$ in $0<x<x_N$, with
$$
x_N^5=\frac{125K\sin\alpha\,V^2t}{16\pi^2\cos^2\alpha}.
$$
The outer front is a discontinuity in this reduced conservation law; the hydrostatic-gradient term and the full local flow resolve the front and apex regions where the outer approximation fails.
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