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Journal of Fluid Mechanics

Drag on a penetrable two-dimensional disk in a superfluid

John Elie Sader, Alexander R. Nunn, Dale I. Pullin

Published Apr 22, 2026• 2 citations
Publisher

Abstract

The motion of a body through a superfluid can generate phenomena distinct from a normal viscous fluid. This includes the absence of drag below a critical body speed with the shedding of quantised vortices and non-zero drag above this speed. These phenomena are often modelled using the Gross–Pitaevskii (GP) equation, which describes the wavefunction of a weakly interacting Bose–Einstein condensate and its superfluid dynamics. We study the drag experienced by a penetrable circular disk of radius, $a$ , in the form of a two-dimensional potential barrier of height, $V_0$ , that is moving in a superfluid of bulk density, $n_\infty$ , by solving the GP equation. The drag is found to exhibit a unimodal dependence on the disk speed for a fixed value of its barrier height. This behaviour is quantified analytically and confirmed using direct numerical solution. The maximum drag per unit length of $F_{\textit{drag}}^{\textit{max}} \approx 5 a n_\infty V_0$ occurs when the barrier height coincides with the relative kinetic energy of the fluid particles. Flow excitations are diminished at a high particle kinetic energy with a commensurate reduction in the drag. This generates a saddle-node bifurcation in the dynamics of a moving disk under an applied force. These results advance understanding of the motion of bodies with finite penetrability, which is relevant to laser experiments probing the superfluidity of Bose–Einstein condensates.

DOI
10.1017/jfm.2026.11467

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The motion of a body through a superfluid can generate phenomena distinct from a normal viscous fluid. This includes the absence of drag below a critical body speed with the shedding of quantised vortices and non-zero drag above this speed. These phenomena are often modelled using the Gross–Pitaevskii (GP) equation, which describes the wavefunction of a weakly interacting Bose–Einstein condensate and its superfluid dynamics. We study the drag experienced by a penetrable circular disk of radius, aaa , in the form of a two-dimensional potential barrier of height, V0V_0V0​ , that is moving in a superfluid of bulk density, n∞n_\inftyn∞​ , by solving the GP equation. The drag is found to exhibit a unimodal dependence on the disk speed for a fixed value of its barrier height. This behaviour is quantified analytically and confirmed using direct numerical solution. The maximum drag per unit length of Fdragmax≈5an∞V0F_{\textit{drag}}^{\textit{max}} \approx 5 a n_\infty V_0Fdragmax​≈5an∞​V0​ occurs when the barrier height coincides with the relative kinetic energy of the fluid particles. Flow excitations are diminished at a high particle kinetic energy with a commensurate reduction in the drag. This generates a saddle-node bifurcation in the dynamics of a moving disk under an applied force. These results advance understanding of the motion of bodies with finite penetrability, which is relevant to laser experiments probing the superfluidity of Bose–Einstein condensates.
The phenomenon of superfluidity in open Bose–Einstein condensates (BEC) is analysed numerically and analytically. It is found that a superfluid phase is feasible above the speed of sound, when forces due to inhomogeneous non-equilibrium processes oppose the contributions of homogeneous processes. Furthermore a regime of accelerating impurities can be observed for particular pumping/decay strategies. All findings are derived within the complex Gross–Pitaevskii (GP) theory extended to include creation and annihilation terms. Utilising this framework the effective force acting on an impurity as it moves with velocity v through the open condensate can be calculated. The result shows that the drag force is continuously increasing with increasing velocity starting from the state of zero motion at v = 0, a property that can be traced down to the additional homogeneous annihilation/creation term in the extended GP model. For very large velocities however we observe a reversion of the drag force. Our findings stand in stark contrast to the concept of a topological phase transition to frictionless flow below a critical velocity as observed for equilibrium BEC analytically (Astrakharchik and Pitaevskii 2004 Phys. Rev. A 70 013608; Pinsker 2017 Physica B 521 36–42), numerically (Winiecki et al 1999 Phys. Rev. Lett. 82 26) and for trapped atoms experimentally (Desbuquois et al 2012 Nat. Phys. 8 645; Zwierlein et al 2006 Nature 442 54–58).
It has been proposed in recent years that an impurity moving in a superfluid will experience drag forces even below the Landau critical velocity [1–3]. Quantum fluctuations that are present in superfluid systems at zero temperature can scatter from impurities present in the fluid. If the impurity is moving, there will be an asymmetric Doppler shift associated with the scattered waves, which will result in a net drag force. This challenges the long-accepted notion of dissipationless flow in superfluids. Analytical estimates of this drag force in superfluid Bose-Einstein condensates using Bogoliubov methods have found a non-zero drag force, resulting from the most sensible choice of boundary conditions in an infinite system. Through stochastic simulations of the 1D Bose gas under the truncated Wigner formalism, we verify the existence of this sub-critical drag force and quantify the drag forces present. We will also determine whether dissipationless flow can be recovered in finite sized systems.
Abstract We study the hydrodynamic forces acting on a small impurity moving in a two-dimensional Bose–Einstein condensate at non-zero temperature. The condensate is modelled by the damped-Gross Pitaevskii (dGPE) equation and the impurity by a Gaussian repulsive potential coupled to the condensate. For weak coupling, we obtain analytical expressions for the forces acting on the impurity, and compare them with those computed through direct numerical simulations of the dGPE and with the corresponding expressions for classical forces. For non-steady flows, there is a time-dependent force dominated by inertial effects and which has a correspondence in the Maxey–Riley theory for particles in classical fluids. In the steady-state regime, the force is dominated by a self-induced drag. Unlike at zero temperature, where the drag force vanishes below a critical velocity, at low temperatures the impurity experiences a net drag even at small velocities, as a consequence of the energy dissipation through interactions of the condensate with the thermal cloud. This dissipative force due to thermal drag is similar to the classical Stokes’ drag. There is still a critical velocity above which steady-state drag is dominated by acoustic excitations and behaves non-monotonically with impurity’s speed.
This thesis examines the manifestation of the critical velocity for superfluidity in dilute gas Bose-Einstein condensates (BECs). Superfluidity is a macroscopic quantum phenomenon characterised by unique properties such as frictionless flow, high thermal conductivity and the presence of quantised vortices. Our study focuses on what is arguably the most well-known attribute: the ability for an object to move through a superfluid without experiencing a drag force when travelling below a threshold critical velocity.The Landau criterion for superfluidity predicts the critical velocity in a BEC is equal to the speed of sound, and this conclusion is supported by a large body of theoretical work. However, a number of experiments have observed lower critical velocities than theory predicts. In the first part of this thesis, we numerically model experiments on the superfluid critical velocity of an elongated, harmonically trapped Bose-Einstein condensate as reported by [P. Engels and C. Atherton, Phys. Rev. Lett. 99, 160405 (2007)]. These experiments swept a repulsive obstacle, formed by a optical dipole potential, through the long axis of a condensate at constant velocity. Their results found an increase in the resulting density fluctuations of the condensate above an obstacle velocity of v _ 0:3 mm s
The problem of the transcritical flow of a Bose-Einstein condensate through a wide repulsive penetrable barrier is studied analytically using the combination of the locally steady ``hydraulic'' solution of the one-dimensional Gross-Pitaevskii equation and the solutions of the Whitham modulation equations describing the resolution of the upstream and downstream discontinuities through dispersive shocks. It is shown that within the physically reasonable range of parameters, the downstream dispersive shock is attached to the barrier and effectively represents the train of very slow dark solitons, which can be observed in experiments. The rate of the soliton emission, the amplitudes of the solitons in the train, and the drag force are determined in terms of the Bose-Einstein condensate oncoming flow velocity and the strength of the potential barrier. Good agreement with direct numerical solutions is demonstrated. Connection with recent experiments is discussed.
The existence of frictionless flow below a critical velocity for obstacles moving in a superfluid is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a nonzero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytical solutions to the Bogoliubov-de Gennes equations for a gray soliton. Our results raise questions regarding the quantum dynamics in the formation of persistent currents in superfluids.