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Physical Review Fluids

Superflow passing over a rough surface: Vortex nucleation

Thomas Frisch, Sergey Nazarenko, Sergio Rica

Published Feb 26, 2024• 6 citations
Publisher

Abstract

The dynamics of a superfluid over a surface exhibits significant differences from compressible flow in ordinary fluids. In ordinary fluids, when the local speed exceeds the sound speed, intrinsic dissipation due to viscosity can enable a shock wave. However, in superfluids, lack of dissipation prevents shock waves. Instead, spatial modulation of a wave train of solitons, as in the figure, allows for a smooth transonic transition. This wave train has been observed to eventually become unstable, leading to quantized vortices and an effective drag on a rough surface. The wave train occurs beneath a lambda-shaped structure with a fore and back-front, reminiscent of ordinary compressible fluids.

DOI
10.1103/physrevfluids.9.024701

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The dynamics of a superfluid over a surface exhibits significant differences from compressible flow in ordinary fluids. In ordinary fluids, when the local speed exceeds the sound speed, intrinsic dissipation due to viscosity can enable a shock wave. However, in superfluids, lack of dissipation prevents shock waves. Instead, spatial modulation of a wave train of solitons, as in the figure, allows for a smooth transonic transition. This wave train has been observed to eventually become unstable, leading to quantized vortices and an effective drag on a rough surface. The wave train occurs beneath a lambda-shaped structure with a fore and back-front, reminiscent of ordinary compressible fluids.
We report on numerical solutions of the Gross-Pitaevskii equation for two-dimensional flow of a superfluid condensate through a small orifice. Above a critical velocity of about 30%30\%30% of the speed of sound, cavitation occurs in the throat of the orifice. The cavitating bubbles form the cores of singly quantized vortices which detach from the boundary and are convected downstream.
Superfluids are extraordinary liquids whose viscosity is zero. This enables them to flow without friction and to pass effortlessly through fine pores or tiny openings (in earthenware pottery for example) that are quite impervious to ordinary fluids such as water. Similarly, an object moving in a superfluid experiences no drag and behaves dynamically as if it were moving in a vacuum. But what will happen if this object is subjected to a constant force? Physical intuition suggests that the object will accelerate uniformly, in the absence of drag, but it is also obvious that this cannot continue indefinitely. Sooner or later, something must surely happen to stop the acceleration (figure 1). The nature of this catastrophic event, in which the superfluidity breaks down at least temporarily, is of considerable intrinsic interest and also sheds much light on the fundamental character of the superfluid state, and the physics of the superfluid under consideration.
An object can move through a superfluid without resistance provided that its speed is below a critical value; higher speeds result in the formation of vortices, which are small volumes of normal fluid and superfluid that rotate and dissipate energy. Although the dynamics of these vortices are considered to be crucial in understanding superfluidity, the strong and complex interactions of the superfluid, dissipative forces, and surface effects in most superfluids (such as liquid helium) all combine to obscure observations.
The dynamics of quantized vortices in weakly interacting superfluids are often modeled by a nonlinear Schrödinger equation. In contrast, we show that quantized vortices in fact obey a non-Hamiltonian evolution equation, which enhances dispersion along the vortex line while introducing a gain mechanism. This allows the vortex medium to support a helical shock front propagating ahead of a dissipative soliton. This dynamic relaxes localized curvature events into Kelvin wave packets. Consequently, a beyond local induction model provides a pathway for decay in low-temperature quantum turbulence.
Superfluid vortex dynamics on an infinite cylinder differs significantly from that on a plane. The requirement that a condensate wave function be single valued upon once encircling the cylinder means that such a single vortex cannot remain stationary. Instead, it acquires one of a series of quantized translational velocities around the circumference, the simplest being $\ifmmode\pm\else\textpm\fi{}\ensuremath{\hbar}/(2MR)$, with MMM the mass of the superfluid particles and RRR the radius of the cylinder. A generalization to a finite cylinder automatically includes these quantum-mechanical effects through the pairing of the single vortex and its image in either the top or bottom end of the surface. The dynamics of a single vortex on this surface provides a hydrodynamic analog of Laughlin pumping. The interaction energy for two vortices on an infinite cylinder is proportional to the classical stream function $\ensuremath{\chi}({\mathbit{r}}_{12})$, and it crosses over from logarithmic to linear when the intervortex separation ${\mathbit{r}}_{12}$ becomes larger than the cylinder radius. An appendix summarizes the connection to an earlier study of Ho and Huang for one or more vortices on an infinite cylinder. A second appendix reviews the topologically equivalent planar annulus, where such quantized vortex motion has no offset, but Laughlin pumping may be more accessible to experimental observation.