Hydrodynamics of superfluids confined in blocked rings and wedges

TL;DR

This paper studies the hydrodynamics and non-classical rotational inertia effects of superfluids confined in blocked ring and wedge geometries, revealing small NCRI in rings and velocity divergence in wedges, with implications for supersolid experiments.

Contribution

It introduces a methodology to compute NCRI effects in complex geometries and analyzes the impact of blocking and vortices on superfluid behavior.

Findings

NCRI is small in blocked rings, leading to a large moment of inertia discontinuity.

Velocity diverges at the origin in wedges with angle > π, indicating normal fluid regions or vortices.

Implications for supersolid helium experiments are discussed.

Abstract

Motivated by many recent experimental studies of non-classical rotational inertia (NCRI) in superfluid and supersolid samples, we present a study of the hydrodynamics of a superfluid confined in the two-dimensional region (equivalent to a long cylinder) between two concentric arcs of radii $b$ and $a$ ($b<a$) subtending an angle $\beta$, with $0 \le \beta \le 2\pi$. The case $\beta= 2 \pi$ corresponds to a blocked ring. We discuss the methodology to compute the NCRI effects, and calculate these effects both for small angular velocities, when no vortices are present, and in the presence of a vortex. We find that, for a blocked ring, the NCRI effect is small, and that therefore there will be a large discontinuity in the moment of inertia associated with blocking or unblocking circular paths. For blocked wedges ($b=0$) with $\beta > \pi$, we find an unexpected divergence of the velocity at the origin, which implies the presence of either a region of normal fluid or a vortex for {\it any} nonzero value of the angular velocity. Implications of our results for experiments on "supersolid" behavior in solid $^4{\rm He}$ are discussed. A number of mathematical issues are pointed out and resolved.