= Small-slope breakdown at an inclined-current nose
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, $h\sim(-3\theta h_N^2y)^{1/3}$. 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.
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