On the instability of three-dimensional flows of an ideal incompressible fluid

Abstract

In a recent paper Lifschitz and Hameiri [Phys. Fluids A 34 (1991) 2644] demonstrated how the stability of general three-dimensional steady flows of an ideal incompressible fluid with respect to short wavelength perturbations can be studied via the geometrical optics method. The flow is unstable if for some stream line the transport equation associated with the wave which is advected by the fluid has a sufficiently rapidly growing solution. In the present paper we show that under certain conditions this equation can be solved explicitly. Using this fact we demonstrate that all axisymmetric flows such that the corresponding poloidal velocity has a point of stagnation, particularly all axisymmetric vortex rings, are unstable. We also describe explicitly exponentially growing solutions of the transport equation for three-dimensional flows with stretching.