Instability of a Flow System

Abstract

This chapter is devoted to the definitions of stability and instability for a flow system. Stability and instability are intended as attributes of a solution of a partial differential equation, or system of equations, governing a flow phenomenon of any kind. The presentation of these concepts starts from a mechanical system, its equations of motion and its phase space. In this context, evolution in time is identified with trajectories in the phase space which can be stable or unstable according to Lyapunov’s definition. Particular trajectories in phase space are the equilibrium states. For equilibrium states, stability may manifest under the form of asymptotic stability. In the case of a flow system, equilibrium states are stationary solutions of the governing equations. The example of a flow system based on a nonlinear Burgers partial differential equation is discussed. This simple system provides the arena where the concepts of convective and absolute instabilities are examined. Other cases are examined relative to channelised Burgers flow and to a convective Cahn–Hilliard model.