Motion and stability of vortices in a finite channel: Application to liquid helium II

Abstract

The paper analyzes the motion of a vortex ring in a coaxial cylindrical cavity, the motion of a vortex pair in a finite channel, the stability of a vortex near a plane wall, and the application of these results to the critical velocities of liquid helium II. In discussing the vortex ring a new representation for the stream function in terms of the modified Bessel functions is given, and this is shown to be equivalent to the usual representations. The new representation enables one to solve for the effect of the boundary. The exact results are suitably approximated when the radius r 0 of the ring is very small or almost the same as the radius R of the cavity. The kinetic energy T increases with r 0 at first, reaches a maximum when r 0 R is nearly 0.9 and then falls rapidly to a small finite value when the vortex touches the wall. The translational velocity V decreases as r 0 increases, becomes zero at r 0 R ∼ 0.9 and then increases in the opposite direction. The two-dimensional case of the vortex pair in a finite channel shows very similar features, and in the limit of the vortex touching the wall the two cases become identical. The straight vortex near a plane wall is shown to be unstable for disturbances along its length under certain conditions. The use of the correct limiting value of the kinetic energy and the usual impulse of the vortex ring in the Landau criterion for the destruction of superflow leads to v s,cD = ( h 2πm )(2ln2 + 1 2 ); D = 2R But the momentum of the vortex ring is not a clearly defined concept and so the Landau criterion is formulated as v c ≥ | T ( ∂T ∂V ) min The results are then nearly the same as the crude Feynman relation v s,cD = ( h 2πm ) · 2ln + ( D 2a ) on account of an approximate cancellation of the effect of the boundary on the energy and the “momentum” of the excitation. It is concluded from the analysis of the experimental observations that the suggested mechanisms give an order of magnitude estimate of the critical velocities, but are incomplete in some respects.