Gravity-driven thin film on an incline

For a film of thickness $h(x,t)$ flowing down a plane inclined by $\alpha$, lubrication theory gives the fluxwhen surface tension is neglected. Mass conservation gives the thin-film equation $h_t+q_x=0$.

Let $x$ be slant distance from an upward apex, and let the surface slope down at angle $\alpha$ to the horizontal. Axisymmetric lubrication theory givesFor conserved volume $V$ and $x_N^3\gg V/\sin \alpha$, the leading similarity solution has $h=\sqrt{x/(5K\sin \alpha t)}$ in $0<x<x_N$, withThe 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.

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 $C{h}^3$.