Energy of a Vortex Ring in a Tube and Critical Velocities in Liquid Helium II

Abstract

Various authors have suggested that critical velocities $v_c$ in liquid helium II may result from the formation of vortex rings according to Landau's criterion, $v_c={\left(\frac{E}{p}\right)}_{min}$, where $E$ is the energy of the ring and $p$ its impulse. In considering the possible formation of rings inside the channel from this point of view, however, the effect of the walls on $E$ and $p$ has been neglected. By solving Laplace's equation in series, we have evaluated the energy of a circular classical vortex ring with an empty streamlined core confined coaxially in a long circular tube of radius $R$; numerical results are presented for various core radii $a$ and ring radii $r_0$. $E$ has a maximum at $r_0\approx 0.9R$, and approaches zero as $r_0\to R$. Boundaries do not affect the impulse, so Landau's criterion applied to such a classical vortex ring gives $v_c=0\mathrm{}$, contradicting experiment. We may conclude that for some reason vortex rings must not be formed inside the channel, unless some special mechanism prevents their formation (or their causing friction if formed) too near the walls. Numerical results are also presented for the exact solution in an unbounded fluid.