Stability of a film flowing down an inclined corrugated plate: The direct Navier-Stokes computations and Floquet theory

Abstract

The paper is devoted to a theoretical analysis of the linear stability of the viscous liquid film flowing down an inclined wavy surface. The study is based on the Navier-Stokes equations in their full statement. The developed numerical algorithm allows us to compute both the steady state solution of the nonlinear equations and the rates of growing or damping in time of the arbitrary two-dimensional disturbances of the solution which are bounded in space. The wall corrugations have a great influence on the disturbances behaviour. There is a critical Reynolds number Recr when the steady-state viscous flow over an undulated surface becomes unstable. It is found that the value of Recr depends essentially both on the topography parameters and the liquid's physical properties. In the case of the flat plate, the critical Reynolds number depends only on the value of the inclination angle. For different values of the Kapitza number, the inclination angle, and the Reynolds number we obtained the regions of the corrugation's parameters (amplitude and period) where all two-dimensional disturbances decay in time.