On a variational principle for the steady flows of perfect fluids and its application to problems of non-linear stability

Abstract

One shows that a steady fluid flow has an extremal energy value among “isovorticed flows”. If the extremum is maximum or minimum, the steady flow is stable with respect to a finite perturbation. In order to clarify the nature of the extremum, an explicit expression for the second variation of energy is given. This way one obtains sufficient conditions of flow stability. These conditions are close to be necessary ones, at least for planar flows.