The stability of fluid motion

Abstract 1. The following notes on the stability of fluid motion arose from a desire to use the energy method, introduced by Reynolds and modified by Orr, as a measure of the comparative degree of stability of various types of flow under different boundary conditions. A few examples are worked out to illustrate this point of view: in 5 a case which resembles the flow of a stream with a free surface; in 7 flow which approximates to a uniform stream between fixed walls without slipping at the walls; in 6,8 motion with other boundary conditions. Before proceeding to these, it seems desirable to give a short account of the method in the form in which it is used later, together with some remarks on its relation to the classical method of small vibrations. 2. We shall consider only two-dimensional motion of an incompressible viscous fluid limited by the planes y = ±a. Let the steady state under an assigned foreive and given boundary conditions be specified by a velocity, U, parallel to the axis of x. Let the disturbed state have velocity components (U + u, v) and let the additional pressure be p. Then, by taking the difference of the two sets of hydrodynamical equations for the two states and neglecting squares and products of the additional velocities, we have ∂u/∂t+U ∂u/∂x+v∂U ∂y= -1/p ∂p/∂x + µ/p ∇2u, ∂v/∂t + U∂v ∂x = -1/p ∂p ∂y + µ/p ∇2v, together with the equation of continuity.