Instability of power-law fluid flow down a porous incline

In this paper we investigate the generation and structure of roll waves developing on the surface of a power-law fluid layer flowing down a porous incline. The unsteady equations of motion for the power-law fluid layer are depth integrated according to the von Kármán momentum integral method accounting for the variation of the velocity distribution with depth. The slip boundary condition at the interface between the fluid layer and the porous plane is based on the assumption that the flow through the porous medium is governed by the modified Darcy’s law, and that the characteristic length scale of the pore space is much smaller than the depth of the fluid layer above. An analytical theory of permanent roll waves is employed to determine under what flow conditions roll waves can exist and to calculate the wavelength, wave height, and speed of the roll waves. The nonlinear stability analysis is also carried out by numerically solving the time dependent governing equations and calculating the nonlinear evolution of infinitesimal disturbances imposed on the uniform and steady flow. Conclusions are drawn regarding the effect of the permeability of the porous medium and flow conditions on the development and characteristics of roll waves arising from the instability of the uniform flow.