Finite-volume gravity-driven thin-film current

Fora current of conserved area $V$ has the first-kind similarity solutionterminated by a shock atIts front height is $h_N(t)=(V/(2Ct){)}^{1/3}$.

The scaling $x\sim C{h}^{2}t$ and $V\sim hx$ gives $V\sim C{h}^{3}t$. Maintaining an asymptotically constant front height therefore requires constant volume flux of order $C{h}^3$.