Instability in gravity-driven flow over uneven surfaces

We consider the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography and surface tension on the stability of the flow. The equations of motion are approximations to the Navier-Stokes equations which exploit the assumed relative shallowness of the fluid layer. Included in these equations are diffusive terms that are second order relative to the shallowness parameter. These terms, while small in magnitude, represent an important dependence of the flow dynamics on the variation in bottom topography and play a significant role in theoretically capturing important aspects of the flow. Some of the second-order terms include normal shear contributions, while others lead to a nonhydrostatic pressure distribution. The explicit dependence on the cross-stream coordinate is eliminated from the equations of motion by means of a weighted residual approach. The resulting mathematical formulation constitutes an extension of the modified integral-boundary-layer equations proposed by Ruyer-Quil and Manneville [Eur. Phys. J. B 15, 357 (2000)] for flows over even surfaces to flows over variable topography. A linear stability analysis of the steady flow is carried out by taking advantage of Floquet-Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed equilibrium flow. The simulations are used to confirm the analytical predictions and to investigate the interfacial wave structure. The bottom profile considered in this investigation corresponds to periodic undulations characterized by measures of wavelength and amplitude. Conclusions are drawn on the combined effect of bottom topography and surface tension.