Modulation Equations for Roll Waves of a liquid film Down an Inclined Plane as a Power-Law Fluid

Roll waves of finite amplitude on a thin layer of non-Newtonian fluid modeled as a power-law fluid are considered. In the long wave approximation, the flow is governed by a non-homogeneous hyperbolic system of equations. As the linearized instability analysis of a uniform flow delivers only a diagnosis of instability, the nonlinear stability is investigated and the criterion for roll waves based on the hyperbolicity of the modulation equation is suggested. The main problem in defining the roll wave stability region on a roll wave diagram is due to the singularities of functions for the mean values and their derivatives near the boundaries of roll wave existence. Asymptotic formulae for nonlinear stability of roll waves of small and maximal amplitudes are derived. Numerical calculation reveals that for a Newtonian fluid, as the bottom inclination decreases downwardly the amplitude of admissible waves diminishes, and the stability domain reduces until it disappears. These results remain valid for a slightly non-Newtonian fluid. For highly non-Newtonian fluid, an inversion in the nature of stability is observed.