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

TL;DR

This paper derives and validates consistent shallow water models for thin power law fluid films on inclined planes, accounting for momentum diffusion and analyzing stability under large-scale perturbations, especially near singular viscosity conditions.

Contribution

It introduces a new formulation of shallow water equations for power law fluids that handles singularities without regularization, valid for flow behavior index n > 1/2.

Findings

Models match Orr-Sommerfeld equations for large-scale perturbations.

Singularity at free surface is effectively managed without regularization.

Flow behavior index n > 1/2 is critical for model validity.

Abstract

In this paper we derive consistent shallow water equations for 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 dynamic of the thin film flows when instabilities like roll-waves arise. These models are validated through a comparison with Orr Sommerfeld equations for large scale perturbations. We only consider 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 thin film and its relation with long wave asymptotic 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 Cauchy momentum equations. No regularization procedure is needed nor distinction between shear thinning/thickening cases. Our analysis is only valid when the flow behavior index $n$ is larger than 1/2 and strongly suggests that the equations are ill posed if $n<1/2$.