A modified Shkadov's model for thin film flow of a power law fluid over an inclined surface

Abstract

A new evolution equation coherent up to order one in the long wave parameter is derived to describe the non-linear behavior of a thin film flow down an inclined plane of a power law fluid for small to moderate Reynolds numbers. The method we have used combines the lubrication theory and the weighted residual approach, with a suitable weighting function. That approach was first developed by Ruyer-Quil and Manneville (2000) for Newtonian fluids. The model has the advantages of both the Shkadov type approach far from criticality and that of Benney close to criticality. To cite this article: M.

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

“…Up to now, there have been only two attempts to derive higher-order shallow-water models for thin power-law films than the first-order models found in Amaouche et al (2009) andFernandez-Nieto et al (2010). In Ruyer-Quil et al (2012), the models are consistent up to order one for inertial terms and up to order two for viscous diffusion terms.…”

“…A first-order consistent shallow-water model was derived in Amaouche, Djema & Bourdache (2009), by using the method of weighted residuals (Ruyer-Quil & Manneville 1998). The first-order models obtained in this way are not unique and moreover are not conservative, which can cause trouble in the presence of shocks.…”

A Resonant Gravity-Driven Flow of a Power-Law Fluid Over a Slippery Topography Substrate

“…In the framework of the integral boundary layer, the surface waves on a fi lm of a power-law fl uid were investigated by Dandapat and Mukhopadhyay [5]. Recently, Amaouche et al [1] have developed an extension of the model equations derived by Ruyer-Quil and Manneville [16] which correctly predict the linear stability threshold. Th e porosity of the wavy bottom has an eff ect on the behavior of a thin liquid fi lm, where a destabilizing infl uence has been deduced by Th iele et al [19], especially the existence of a jump boundary condition.…”