On Green's functions for small disturbances of plane Couette flow

Abstract

The linearized stability of plane Couette flow is investigated here, without using the Orr–Sommerfeld equation. Rather, an unusual symmetry of the problem is exploited to obtain a complete set of modes for perturbations of the unbounded (no walls) flow. An explicit Green's function is constructed from these modes. The unbounded flow is shown to be rigorously stable. The bounded case (with walls) is investigated by using a ‘method of images’ with the unbounded Green's function; the stability problem in this form reduces to an algebraic characteristic equation (not a differential-equation eigenvalue problem), involving transcendental functions defined by integral representations.