Modulation Equations for Roll Waves on Vertically Falling Films of a Power-Law Fluid

Abstract

Waves of finite amplitude on a thin layer of non-Newtonian fluid modelled as a power-law fluid are considered. In the long wave approximation, the system of equations taking into account the viscous and nonlinear effects has the hyper- bolic type. For the two-parameter family of periodic waves in the film flow on a vertical wall the modulation equations for nonlinear wave trains are derived and investigated. The stability criterium for roll waves based on the hyperbolicity of the modulation equations is suggested. It is shown that the evolution of stable roll waves can be described by self-similar solutions of the modulation equations

Formation and evolution of roll waves in a shallow free surface flow of a power-law fluid down an inclined plane

“…The stability of two-layer flows in open and closed channels, as well as in a Hele-Shaw cell, is considered in [6,3] and [12], where the possibility of the roll waves formation in such systems is shown. In [4,5] the modulation equations for free surface shallow flows are derived and the stability criterion of roll waves is formulated in terms of hyperbolicity of these equations. Starting with an expansion of the velocity field in terms of polynomial test functions and using a weighted-residual technique based on a classical long-wavelength expansion more complex depth-averaged models for Newtonian and power-law fluid are derived and verified in [27,26].…”