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

Abstract

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 regularization 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.

Inverse rheometry and basal properties inference for pseudoplastic geophysical flows

“…at vanishing shear-rate) in the power-law shear-thinning constitutive law, in an inverse context, relying on surface velocity observations. As a matter of fact, this singularity might lead to an ill-posed direct problem if a free-surface is considered as pointed out in [16]. For different sliding regimes, sensitivity analysis demonstrate that the refinement of the mesh close to the surface, while leading to the appearance of a stiff (very high viscosity) layer, does not affect the solution and that the viscosity singularity remains passive in the model confirming numerically the analysis done in [17].…”

“…But, this law leads to an infinite viscosity singularity for vanishing shear-rates (in the shear-thinning case, n > 1, see equation (6)). In other respect, it has been demonstrated that this singularity might lead to an ill-posed direct problem if a free-surface is considered (see [16]).…”

Stability of a flow down an incline with respect to two-dimensional and three-dimensional disturbances for Newtonian and non-Newtonian fluids

“…Then, using a long-wave approximation, they demonstrated a linear evolution of the critical Reynolds number as a function of both cotγ and the power-law exponent n. This study, however, is limited by the singularity introduced by the viscosity law in the model: a power-law describes an infinite viscosity at the free surface, which is not physically consistent. To remove this singularity, some authors considered a regularized power-law model: Ruyer-Quil et al [5] by introducing a Newtonian plateau at small strain rate and Noble and Vila [6] by introducing a weaker formulation of the Cauchy momentum equations. They have shown the relevant influence of shear-thinning properties on the primary instability.…”

Asymptotic expansion of the velocity field within the front of viscoplastic surges: comparison with experiments

“…Lubrication models typically reduce to a single advection-diffusion equation for the evolution of flow height h (Benney 1966;Hunt 1994;Huppert 2006;Ancey, Cochard & Andreini 2009). Shallow-water models are generally formulated as systems of two coupled equations (in two dimensions) for flow height h and discharge q or depth-averaged velocitȳ u = q/h (Savage & Hutter 1989;Ancey et al 2007;Iverson 2013;Noble & Vila 2013), although recently more robust three-equation models, including an additional enstrophy variable, have also been proposed (Richard, Ruyer-Quil & Vila 2016). In these models, fluid rheology is accounted for through closure terms that implicitly depend on the distribution of longitudinal velocity u(y) (with y the cross-stream coordinate) across the flowing layer (Ng & Mei 1994;Ruyer-Quil & Manneville 2000;Hogg & Pritchard 2004).…”

“…Lubrication models proceed from a leading-order expansion (Benney 1966;Ruyer-Quil & Manneville 1998). At the next level of approximation, derivation of consistent shallow-water models requires expansion of the velocity profile u(y) up to at least O( ) terms (Ruyer-Quil & Manneville 2000;Noble & Vila 2013). Yet, most shallow-water models used in practical applications, either for Newtonian of non-Newtonian fluids, are based on simpler, ad-hoc assumptions.…”