Stability Analysis of a Falling Film Flow Down a Plane with Sinusoidal Corrugations

The liquid viscous film falling down a vertical wall with sinusoidal relief is considered. The linear stability of steady-state flow with respect to time-periodic disturbances is studied using the Floquet theory. It is shown that in the case of applying corrugations the variation in the disturbance growth rate is proportional to the second power of their undulations. Depending on the relief parameters there exist two possibilities: the instability domain can expand or certain disturbances can be stabilized. The growth rates are obtained numerically and analytically in the approximation of low-amplitude corrugations. The development of waves from small disturbances is simulated within the framework of nonlinear equations and the formation of structures whose wavelength is significantly greater than the space relief period is found out.