Travelling front of a gravity-driven thin film

For $h_t={\partial }_x(-A{h}^3+B{h}^3{h}_x)$, $A,B>0$, a front $h=F(x-ct)$ joining $h_0$ behind it to zero at its tip has $c=A{h}_0^2$. Integrating once gives $F^{\prime }=(A/B)(1-h_0^2/F^2)$ andNear 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.

For a gravity-current nose of height $h_N$ on a small incline of slope $\theta$, the travelling front of a gravity-driven thin film satisfies $h_y=\theta (1-h_N^2/h^2)$. Close to its dry tip, $h\sim (-3\theta h_N^{2}y{)}^{1/3}$. The slope becomes order one at $h\sim \sqrt{\theta }{h}_N$, giving an $O(\sqrt{\theta }{h}_N)$ region where lubrication theory fails. The main nose length is $h_N/\theta$. Comparing capillary pressure gradients $\gamma h_N/(h_N/\theta {)}^3$ with gravitational forcing $\rho g\theta$ gives the crossover $h_N\sim \theta \sqrt{\gamma /(\rho g)}$. This concerns the main nose length; capillarity may already matter in a much smaller tip region.