An Exact Solution of the Orr‐Sommerfeld Equation for Plane Couette Flow

Abstract

The equation which governs the stability of plane Couette flow admits solutions of either well‐balanced or dominant‐recessive type. As is well known, the solutions of well‐balanced type can be expressed in terms of simple exponential or hyperbolic functions. The main aim of this paper, therefore, is to show that the solutions of dominant‐recessive type can be expressed exactly as the sum of three products which involve certain rapidly and slowly varying functions. The rapidly varying functions are simply Airy functions, together with their first integrals and first derivatives. Of the three slowly varying functions, one is expressible in terms of a hyperbolic function and the other two have simple integral representations which, for bounded values of the wavenumber α, have convergent expansions in powers of (iαR)−1. An application of these results is also made to the problem of semibounded plane Couette flow, and it is shown that the eigenvalue relation for this problem can be expressed in an extremely simple form.