Weakly Nonlinear Instability
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In the preceding chapter, we examined the stability of a wide range of flows, most of them steady solutions to their governing fluid equations. In every case, our approach was to perturb the steady flow in order to define a new problem for the evolution of the perturbations themselves, or in other words the robustness of the steady base flow. We then drew conclusions about the stability properties of the original flow based on the evolution of the perturbations. To make these problems tractable, we assumed that the perturbations were small, neglecting all nonlinear terms, that is, terms involving products of the small perturbation quantities. As a linear instability grows, it will eventually become large enough in magnitude that nonlinear terms can no longer be neglected. In this case, nonlinearity plays several roles, but their main role, and of greatest interest to us, is the ability to saturate the exponential growth.