Finite-volume gravity-driven thin-film current
= Finite-volume gravity-driven thin-film current
For
$$
h_t+(Ch^3)_x=0,\qquad C>0,
$$
a current of conserved area $V$ has the first-kind similarity solution
$$
h(x,t)=\sqrt{\frac{x}{3Ct}},
\qquad 0<x<x_N(t),
$$
terminated by a shock at
$$
x_N(t)=\left(\frac{27CV^2t}{4}\right)^{1/3}.
$$
Its front height is $h_N(t)=(V/(2Ct))^{1/3}$.