NONLINEAR INSTABILITY OF VISCOUS PLANE COUETTE-FLOW .1. ANALYTICAL APPROACH TO A NECESSARY CONDITION

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 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 epsilon, the non-dimesnsional amplitude of the defect. For large enough epsilon-R (R being the Reynolds number based on the width of the defect), the maximum growth rate is about 1/2-epsilon, at approximately half the 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 epsilon is greater than R(L)-1/3, 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 [(epsilon-R(L))-1/2, epsilon].