Hydrodynamics of superfluids confined in blocked rings and wedges

Motivated by many recent experimental studies of nonclassical 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 $0⩽\beta ⩽2\pi$. 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 any nonzero value of the angular velocity. Implications of our results for experiments on “supersolid” behavior in solid ${}^4\mathrm{He}$ are discussed. A number of mathematical issues are pointed out and resolved.

Figure 1

(Color online) The velocity field for a blocked ring with $c=0.5$. The first panel shows the relative strengths of the velocity field as a function of position. The second panel is the radial component (in units of $\Omega a$) plotted vs $\rho \equiv r∕a$ at azimuthal angles $\varphi$ (from bottom to top) of $\pi ∕16$, $\pi ∕8$, $\pi ∕4$, $7\pi ∕4$, $15\pi ∕8$, $31\pi ∕16$. The third panel, in the same units, shows the azimuthal component of the velocity vs $\varphi$ at $\rho =0.6,0.75,0.9$.

Figure 2

Moment of inertia of an obstructed ring in terms of its aspect ratio $c\equiv b∕a$. In the top panel the ratio $R$ of $I_{\mathit{SF}}$ [Eq. (2.18)] to the rigid-body value is plotted, while in the bottom panel we plot $I_{\mathit{SF}}$ itself, in units such that $a=1$. The maxima in the two plots are at different values of $c$.

Figure 3

(Color online) The ratios $I_{\mathit{SF}}∕I_{\mathit{RO}}$ (upper curve) and $I_{\mathit{SF}}^{\mathrm{c.m.}}∕I_{\mathit{RO}}^{\mathrm{c.m.}}$ (lower curve) for a superfluid wedge as a function of the opening angle $\beta$, $0<\beta ⩽2\pi$. $I_{\mathit{SF}}$ is calculated from Eq. (2.24).

Figure 4

(Color online) Plots of the velocity field inside the wedge for $\beta =(7∕8)2\pi$. This should be compared with the first panel of Fig. 1.

Figure 5

(Color online) The ratios $I_{\mathit{SF}}∕I_{\mathit{RO}}$ (upper curve) and $I_{\mathit{SF}}^{\mathrm{c.m.}}∕I_{\mathit{RO}}^{\mathrm{c.m.}}$ (lower curve) for an annular wedge [Eq. (2.31)] plotted as a function of the opening angle $\beta$, $0<\beta ⩽2\pi$, at a fixed value of $c=0.5$.

Figure 6

(Color online) The critical angular velocity for vortex nucleation in a ring $(\beta =2\pi )$. Here the critical value of the parameter $\gamma$ (i.e., ${\Omega }_1{a}^2∕\kappa$) is plotted as a function of $c$. The circles are numerical results, connected by straight dashed lines. The increase at larger $c$ shows that the nucleation of vortices is unfavorable in that case.

Figure 7

(Color online) Fields produced by a nucleated vortex in an obstructed ring with $c=0.5$, at $\Omega ={\Omega }_1$. Only the fields produced by the vortex are included. Its position [marked by a (blue) open dot] is at the optimal value (see text) $r_v∕a=0.74$. The total flow is the sum of that shown in this figure, weighed by a factor of $1∕\gamma$, and that in the top panel of Fig. 1. Because $\gamma$ is rather large, the result would be hard to distinguish from that shown in Fig. 1.