Stability of a liquid flow in a flat channel with wavy walls

A viscous fluid flow between two wavy horizontal surfaces unlimited in longitudinal and transverse directions is considered. The full Navier–Stokes equations are applied to study the linear stability of such a flow with respect to various three-dimensional disturbances. Two types of wall waviness are studied: longitudinal and transverse periodic corrugation. At the first stage, the main solution is obtained and the initial equations are linearized in the vicinity of this solution. At the second stage, the generalized problem of determining eigenvalues is solved and the entire possible spectrum of disturbances is analyzed. The varied parameters are the Reynolds number, amplitude, period, and shape of the corrugation. Disturbances of velocity and pressure fields are generally characterized by two wave numbers, which are additional parameters. The influence of the parameters and shape of the wall waviness on the region where the laminar-turbulent transition begins is investigated.