Remarks on the behaviour of flow of a three-dimensional perfect fluid in the presence of a small perturbation of the initial velocity field

Abstract

Euler equations of the three-dimensional motion of a perfect incompressible fluid, linearized for a nearly stationary flow are considered and the class of stationary flows for which these linearized equations admit exact explicit solutions is indicated. The analysis of derived equations shows that in some stationary flows the perturbation buildup considerably differs from that obtaining in cases generally considered in the theory of hydrodynamic stability: there appears an infinitely great number of unstable configurations, the flow pattern is difficult to predict (since an approximate determination of perturbation development with time necessitates a rapidly increasing amount of information about initial conditions, etc). These differences are due to the different geometry of stationary flows. In the recently constructed models of stationary flows the assumption is made that a fluid particle in motion stretches into a filament or ribbon whose length exponentially increases with time, while in the usually considered flows the length is assumed to be a linear function of time. In two-dimensional flows the phenomenon of exponential stretching of particles is impossible. It is shown that this is, also, impossible in three-dimensional flows in which the vectors of velocity and viscosity are not collinear.