Instability of steady flows with constant vorticity in vessels of elliptic cross-section

Abstract

The problem of the stability of steady flows of a perfect incompressible fluid in vessels of elliptic cross-section is studied. The flow velocity field of the main stream is a linear function of the coordinates and the vorticity is constant. The spectral problem for the linear perturbations is solved using the mehtod of consecutive approximations. The instability of the flows to a first approximation is demonstrated. A special case of the flow in a triaxial ellipsoid is analysed in detail. Theoretical predictions agree well with the experimental results /1/. The present paper, unlike the analysis carried out in /1/, deals with an appreciably wider class of perturbations and the Galerkin method of rough a priori approximation is not used. The problem of stability of flows of this type is of interest when describing the properties of a liquid-filled gyroscope /2–4/ and the behaviour of the star and planetary cores /5/. At the same time, a flow with a linear velocity field represents the simplest example of the realization of the new mechanism of instability of rotational flows connected with disturbance of the rotational symmetry /6–9/.