Instability in gravity-driven flow over uneven permeable surfaces

We develop a theoretical model for inclined free-surface flow over a porous surface exhibiting periodic undulations. The effect of bottom permeability is incorporated by imposing a slip condition that accounts for the nonplanar geometry of the fluid–porous medium interface. Under the assumption of shallow flow, equations of motion accounting for inertial effects are obtained by retaining in the Navier-Stokes equations terms that are up to second-order with respect to a small shallowness parameter. The explicit dependence on the cross-stream coordinate is eliminated from these equations by means of a weighted residual procedure. A linear stability analysis of the steady flow is performed in connection with Floquet–Bloch theory. The results predict that bottom permeability has a destabilizing influence on the flow. A physical explanation has been proposed which involves examining how permeability affects the steady-state flow. Conclusions are drawn regarding the combined effect of the surface tension of the fluid and the parameters describing the bottom surface including permeability, inclination and the amplitude and wavelength of the undulations that generate the bottom topography. A numerical scheme for solving the fully nonlinear governing equations is also outlined. The instability of particular steady flows is determined by conducting nonlinear simulations of the temporal evolution of the flow and comparisons are made with the predictions from the linear analysis. Comparisons with existing experimental data are also included.