Travelling front of a gravity-driven thin film
= Travelling front of a gravity-driven thin film
For $h_t=\partial_x(-Ah^3+Bh^3h_x)$, $A,B>0$, a front $h=F(x-ct)$ joining $h_0$ behind it to zero at its tip has $c=Ah_0^2$. Integrating once gives $F'=(A/B)(1-h_0^2/F^2)$ and
$$
\xi=\frac BA\left[F+\frac{h_0}{2}\log\frac{h_0-F}{h_0+F}\right].
$$
Near the tip, $F\sim(-3c\xi/B)^{1/3}$. Its unbounded slope signals the failure of the small-slope <lubrication approximation> sufficiently close to the <contact line>.