Energy and Motion of Vortex Rings in Liquid Helium II in the Presence of Various Plane Obstacles

Abstract

A classical calculation is given for the kinetic energy of a vortex ring in an inviscid fluid (He II) in the presence of an infinitely extended plane, a coaxial circular disk, and an infinitely extended plane with a coaxial circular aperture. The energy is obtained as an explicit function of the ring position relative to the obstacle being considered. Formulas are given for the velocity components of the vortex-ring motion, the force exerted on an obstacle by a stationary vortex ring, and the impulse of the vortex ring in the presence of the obstacle. In the case of the circular aperture, there is found to be a critical energy (or ring size) beyond which a vortex ring cannot pass through the aperture. The path of the vortex ring near the different obstacles is obtained by numerical computation, departing from the explicit energy expression. The calculation method is easily extended to other axisymmetric configurations and consists of Fourier and Hankel integral-transform techniques in combination with results from the theory of dual integral equations.