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

Abdelaziz Boudlal,V Yu Liapidevskii

doi:10.4236/wjm.2012.21001

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.

Highlights

Mud flows are frequently encountered in mountainous regions, especially after torrential rains, and often exhibit a series of breaking waves
Based on long wave approximation, Dressler’s theory of roll waves was extended in [1] to a shallow layer of fluid mud, which has been modelled as a power law fluid
It is shown in the previous section that analogously to the roll waves in open channel flows governed by the classic shallow water theory (Whitham [10]), the periodic travelling waves in film flow of a non-Newtonian fluid can be represented by the two-parameter family of discontinuous solutions of (2.5)

Summary

Introduction

Mud flows are frequently encountered in mountainous regions, especially after torrential rains, and often exhibit a series of breaking waves (roll waves). Based on long wave approximation, Dressler’s theory of roll waves was extended in [1] to a shallow layer of fluid mud, which has been modelled as a power law fluid. It has been shown, that if the fluid is highly non-Newtonian, very long waves may still exist even if the corresponding uniform flow is stable to infinitesimal perturbations. All results presented can be regarded as a generalization to a power law mud fluid in laminar flow regime of non linear stability method alredy applied to Newtonian turbulent flows in open channels (Boudlal & Liapidevskii [9])

Governing Equations

Roll Waves

Modulation Equations

Roll Wave Dynamics

12 Re q h2

Conclusion