Instability of Two-Dimensional Unidirectional Shear Flows

An important problem is to analyze the stability of fluid flows. With the exception of the Taylor-Couette and Saffman Taylor problems, this chapter has focused on stability questions when the base state of the system was one with no motion (or rigid-body motion), so that instability addresses the conditions for spontaneous onset of flow. An equally valid question is whether a particular flow, such as Poiseuille flow in a pipe (or any of the other flows that we have analyzed in previous chapters of this book), is stable, especially to infinitesimal perturbations as linear instability determines whether the particular flow is actually realizable in experiments. This question was first mentioned back in Chapter 3 when we analyzed simple unidirectional flow problems and noted that solutions such as Poiseuille s solution for flow through a tube was a valid solution of the Navier-Stokes equations for all Reynolds numbers, even though common experience tells us that beyond some critical Reynolds number there is a transition to turbulent flow in the tube. 

The conclusion from this discussion is that the linear stability of flows is a complicated topic, even if we restrict ourselves to a simple class of problems such as steady, 2D, unidirectional flows, and it is not possible to provide a comprehensive summary in the space available. Instead, we consider only an introduction to the stability for this limited class of problems, with the intention of giving a qualitative sense of the analysis and some very basic results. In particular, we summarize the theory for an inviscid fluid, which has been 

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