Free-surface flows with large slopes: Beyond lubrication theory

The description of free-surface flows can often be simplified to thin-film (or lubrication) equations, when the slopes of the liquid-gas interface are small. Here, we present a long-wavelength theory that remains fully quantitative for steep interface slopes, by expanding about Stokes flow in a wedge. For small capillary numbers, the variations of the interface slope are slow and can be treated perturbatively. This geometry occurs naturally for flows with contact lines: we quantify the difference with ordinary lubrication theory through a numerical example and analytically recover the full Cox-Voinov asymptotic solution.