Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

In this paper we derive consistent shallow-water equations for the flow of thin films of power-law fluids down an incline. These models account for the streamwise diffusion of momentum, which is important to describe accurately the full dynamics of thin-film flows when instabilities such as roll waves arise. These models are validated through a comparison with the Orr-Sommerfeld equations for large-scale perturbations. We consider only laminar flow for which the boundary layer issued from the interaction of the flow with the bottom surface has an influence all over the transverse direction to the flow. In this case the concept itself of a thin film and its relation with long-wave asymptotics leads naturally to flow conditions around a uniform free-surface Poiseuille flow. The apparent viscosity diverges at the free surface, which, in turn, introduces a singularity in the formulation of the Orr-Sommerfeld equations and in the derivation of shallow-water models. We remove this singularity by introducing a weaker formulation of the Cauchy momentum equations. No regularization procedure is needed, nor any distinction between shear thinning and thickening cases. Our analysis, though, is only valid when the flow behaviour index n is larger than 1/2, and strongly suggests that the Cauchy momentum equations are ill-posed if n <= 1/2.