Instability criteria for the flow of an inviscid incompressible fluid

Abstract

We present a geometric estimate from below on the growth rate of a small perturbation of a threedimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for Euler's equations. We use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable.PACS numbers: 47.20.-k In a companion paper Vishik and Friedlander 1 obtain a universal geometric estimate from below on the growth rate of a small perturbation of a three-dimensional steady flow of an inviscid incompressible fluid. In this Letter we discuss certain implications concerning instability for Euler's equations that follow from the existence of this estimate and we demonstrate that the estimate gives effective criteria for local instability. In particular, the existence of a hyperbolic stagnation point implies that the steady flow is unstable. An important feature of our approach which allows us to obtain effective criteria is that, unlike many previous approaches to hydrodynamic stability, we do not study the spectrum but rather we consider the growth rate of the relevant Green's function as t-> °°.There is a very extensive literature concerning the field of hydrodynamic stability (for references see, for example, Drazin and Reid 2 ). We briefly mention some of the work whose techniques are related to those that we employ. Eckhoff and Storesletten 3 and Eckhoff 4 study the stability of azimuthal shear flows of a compressible fluid and more generally symmetric hyperbolic systems using an approach based on the generalized progressing wave expansion. 5,6 Eckhoff shows that local instability problems for hyperbolic systems can be essentially reduced to a local analysis involving ordinary differential equations (ODE) and algebraic equations only. We show that the same conclusion can be drawn for Euler's equations for an ideal fluid. These equations do not form a hyperbolic system; hence several additional technical details arise in the analysis. Bayly 7 studies the stability of quasi-twodimensional steady flows via an analysis of a Floquet system of ODE. He shows that the Floquet exponent gives the growth rate for a family of instabilities which include the Rayleigh centrifugal instability, the LeibovichStewartson columnar instability, and the elliptic vortex instability. We note that the instability criteria that we present in this Letter are equivalent to those of Bayly in the particular case of quasi-two-dimensional steady flows. Lifschitz 8 uses WKB methods to construct part of the continuous spectrum for axisymmetric steady flows. Using methods inspired by magnetohydrodynamics, he obtains a necessary stability condition for a vortex ring with respect to localized three-dimensional perturbations.Let u(x) be a steady solution of Euler's equations governing the motion in 3D of an inviscid incompressible fluid:(1)The 3D vector field u(x) denotes the velocity and the scalar field P(x) denotes the pressure in the fluid. We consider the linear...

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