Gravity-driven ice flow on a conical slope
= Gravity-driven ice flow on a conical slope
For a thin <Newtonian fluid> film on a conical slope at angle $\alpha$, neglecting the thickness-gradient contribution to the driving pressure gives <volume flux per unit width> $q=Dh^3$, where $D=g\sin\alpha/(3\nu)$. Radial <mass conservation> is
$$
h_t+\frac1x\partial_x(xDh^3)=a(x).
$$
The distance $x$ is measured along the slope; the horizontal radius is $x\cos\alpha$, so total volume is $2\pi\cos\alpha\int xh\,dx$.