Linear stability of the viscous liquid flow in a channel with corrugated walls

Using the complete Navier-Stokes equations, we investigate the linear stability of a viscous flow in a channel with various corrugated walls. We consider both the longitudinal corrugations when the main flow has two velocity components and the transversely corrugated wall when the basic flow has one velocity component. In frame of one approach and in a wide range of variations in the corrugation parameters and shapes, we analyze the neutral curves for the linear stability problem. We calculate the critical Reynolds number above which the main flow is unstable and there are disturbances growing in time. Perturbations of the velocity and pressure fields are, in general, three-dimensional with two wave numbers. We solve numerically the generalized eigenvalue problem. In the case of the transversely corrugated wall, we obtain two regions of parameters where the dependencies of the critical Reynolds number on the corrugation parameters are qualitatively different. The limits of these regions depend only on the ratio of the amplitude and period of corrugations. In the case of the longitudinal-corrugations, we find the corrugation parameters where the basic flow is unstable with respect to the periodic disturbances with the same wavelength as the corrugations wavelength and undergoes “the unavoidable transition” to a flow with more complicated behaviour in time.