Numerical and asymptotic results on the linear stability of a thin film spreading down a slope of small inclination

Abstract

We model a thin liquid film moving down a slope using the lubrication approximation with a slip condition. The travelling-wave solution is derived for small inclination angle α, using singular perturbation methods, and compared to the numerical solution. For the linear stability analysis we combine numerical methods with the long-wave approximation and find a small but finite critical α* below which the flow remains linearly stable to spanwise perturbations. This is contrasted with the vanishing of the hump of the travelling-wave solution. Finally, the prevailing linear stability of the travelling-wave at small inclination angles is compared with recent related results using a precursor model. Here, though, a strong dependence on the magnitude of the contact angle is found, which we think has not been observed before.