Nonlinear instability of viscous plane Couette flow Part 1. Analytical approach to a necessary condition

Abstract

We perform a two-dimensional analytical stability analysis of a viscous, unbounded plane Couette flow perturbed by a finite-amplitude defect and generalize the results obtained in the inviscid limit by Lerner and Knobloch. The dispersion relation is derived and is used to establish the condition of marginal stability, as well as the growth rates at different Reynolds numbers. We confirm that instability occurs at wavenumbers of the order of ε, the non-dimensional amplitude of the defect. For large enough εR (R being the Reynolds number based on the width of the defect), the maximum growth rate is about ½ε, at approximately half the critical wavenumber. We formulate the instability conditions in the case where the flow has a finite extension in the downstream direction. Instability appears when ε is greater than R_L1/3, where R_L is the Reynolds number based on the downstream scale, and when the ratio of the defect width to the downstream scale lies in the interval [(εR_L}^−½, ε].