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Journal of Physics A Mathematical and General

Motions in a Bose condensate: VII. Boundary-layer separation

Natalia G. Berloff, Paul Harry Roberts

Published May 24, 2000• 78 citations
Publisher

Abstract

The Bose condensate model is used to analyse the superfluid flow around an ion (modelled as a solid sphere) and to elucidate the mechanism of vortex ring emission from the sphere that occurs if its velocity exceeds a critical value. An asymptotic expansion is developed for the steady subcritical flow, using the ratio of the healing length to the radius of the sphere as a small parameter. This expansion allows for the compressibility of the condensate, and converges well enough for the critical ion velocity to be calculated accurately. The flow for supercritical ion velocities is computed numerically. Particular attention is paid to the question of why the vortex rings are emitted at a preferred location on the sphere's surface.

DOI
10.1088/0305-4470/33/22/307

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The Bose condensate model is used to analyse the superfluid flow around an ion (modelled as a solid sphere) and to elucidate the mechanism of vortex ring emission from the sphere that occurs if its velocity exceeds a critical value. An asymptotic expansion is developed for the steady subcritical flow, using the ratio of the healing length to the radius of the sphere as a small parameter. This expansion allows for the compressibility of the condensate, and converges well enough for the critical ion velocity to be calculated accurately. The flow for supercritical ion velocities is computed numerically. Particular attention is paid to the question of why the vortex rings are emitted at a preferred location on the sphere's surface.
The Bose condensate model is used to elucidate the motion of the electron bubble in superfluids. An asymptotic expansion is developed for steady subcritical flow. Numerical integration of the coupled nonlinear Schrödinger equations, that describe the evolution of the wavefunctions of the Bose condensate and the impurity, is used to study the nucleation and capture of vortex rings. Because an electron bubble is made oblate by its motion relative to the condensate, the critical velocity for the vortex nucleation is reduced by about 20%, in agreement with experiments.
A procedure is given which in principle permits the exact evaluation of any term in the asymptotic expansion in a/c of the velocity U o and energy E of a circular vortex line of radius c in a Bose condensate (where a is the healing length). The procedure is used to obtain the two leading terms (previously only inaccurately determined by a variational calculation), namely U 0 = kappa /4 pi c(In(8c/a)-0.615); E= 1 / 2 rho kappa 2 c(1n(8c/a)-1.615); where kappa denotes the quantum of circulation, and rho denotes the fluid density at infinity.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.
For Pt.I, see abstr. A71357 of 1971. The linearized perturbation equations for the rectilinear and large circular vortex lines in a Bose condensate are studied. An asymptotic matching procedure is given whereby the frequency omega for the (bound state) oscillations of the rectilinear vortex, for wavelengths large compared with the healing length a, is found. A similar asymptotic procedure is employed to investigate the behaviour of the large ring vortex when its circular axis is slightly perturbed. It is found that the vortex, whose radius c is large compared with a, is stable for all small displacements (and executes simple harmonic vibrations).
Nucleation of vortex rings accompanies the collapse of ultrasound bubbles in superfluids. Using the Gross-Pitaevskii equation for a uniform condensate we elucidate the various stages of the collapse of a stationary spherically symmetric bubble and establish conditions necessary for vortex nucleation. The minimum radius of the stationary bubble, whose collapse leads to vortex nucleation, was found to be 28+/-1 healing lengths. The time after which the nucleation becomes possible is determined as a function of the bubble's radius. We show that vortex nucleation takes place in moving bubbles of even smaller radius if the motion makes them sufficiently oblate.
Nucleation of vortex rings accompanies the collapse of ultrasound bubbles in superfluids. Using the Gross-Pitaevskii equation for a uniform condensate we elucidate the various stages of the collapse of a stationary spherically symmetric bubble and establish conditions necessary for vortex nucleation. The minimum radius of the stationary bubble, whose collapse leads to vortex nucleation, was found to be 28+/-1 healing lengths. The time after which the nucleation becomes possible is determined as a function of the bubble's radius. We show that vortex nucleation takes place in moving bubbles of even smaller radius if the motion makes them sufficiently oblate.