Thin power law film flow down an inclined plane: consistent shallow water models and stability under large scale perturbations
Résumé
The aim of this paper is to provide a better understanding of the derivation of two moments approximate models of shallow water type for thin power law films down an incline. We limit ourselves to the case of 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. This Poiseuille flow exhibits a divergence of the apparent viscosity 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 inverting the relation between the fluid strain and the deformation tensor and by keeping the fluid strain as a variable. No regu-larization procedure is needed here, in contrast to RQ [RQ]. In order to derive the shallow water equations, we compute a second order asymptotic expansion of the fluid velocity field and fluid strain in the shallow water regime through a nonlinear iterative scheme. We show that a linearized version of this scheme naturally provides an expansion of the eigenfunctions of the Orr-Sommerfeld equations in the small wavenumber/small frequency regime. The shallow water 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.
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