9.1 Linear Stability Theory of Fluid Flows

Chapters 3 and 4 examined several examples that involved the stability of inviscid flows, such as the round jet, the vortex street, and several cases of interfacial waves. In those cases an infinitesimal disturbance was added to a primary flow, and then it was determined whether there was a possibility of the disturbance growing, decaying, or remaining unchanged. The driving mechanisms involved in those examples were gravity and surface tension interacting with the momentum of the flow.

In this chapter the same type of analysis is used and applied to laminar viscous flows as well. The presence of viscosity in some cases will dampen the destabilizing influences and in others act to enhance them and thus destabilize the primary flow. Flows frequently are laminar for sufficiently small values of a dimensionless parameter, such as the Reynolds number, and become turbulent once the parameter becomes sufficiently large. While this transition value of the parameter can be found experimentally, as Reynolds did for pipe flow, in many cases a flow has so many describing parameters that a complete experimental study becomes too costly. An analytic approach then can be desirable, with the added advantage of possibly shedding some light on the physical mechanisms involved in the transition.

The classical approach in studying such transitions, introduced in the nineteenth century in the study of inviscid flows, is to take the solution for the laminar flow and add to it a very small disturbance. There is usually...