The article considers the dramatic phenomenon of seemingly frictionless flow of slow-moving superfluids. Specifically the question of whether an object in a superfluid flow experiences any drag force is addressed. A brief account is given of the history of this problem and it is argued that recent advances in ultracold atomic physics can shed much new light on this problem. The article presents the commonly held notion that sufficiently slow-moving superfluids can flow without drag and also discusses research suggesting that scattering quantum fluctuations might cause drag in a superfluid moving at any speed.
Actually, the fundamental feature of Bose condensation is the macroscopic occupation of a single quantum state, whatever that state may be. As long as such a macroscopically-occupied state exists, the system will exhibit superfluid behavior. Superfluid flow thus appears as a "collective" phenomenon, in which the particles move together in order to preserve the macroscopic occupation of a single state. The equilibrium is metastable, because obviously there exists a state of lower energy, that of the fluid at rest; the essential point is that no transitions to the latter state will take place. Metastable superfluid flow is only possible because it involves condensate motion. The superfluid velocity is thus proportional to the gradient of the phase of the condensate wave-function. The superfluid motion is thus a uniform translation, in contrast with ordinary viscous flow, for which the fluid velocity varies from the walls to the center of the tube.
We consider the dynamics of a supersolid state that is based on Leggett's idea of local phase flow around the atomic sites of the solid. This leads to a local microscopic superfluid velocity that is not Galilean and is defined only relative to the solid and a
T=0 superfluid fraction ${\ensuremath{\rho}}_{S}/\ensuremath{\rho}<1$ (with $\ensuremath{\rho}$ being the macroscopic mass density). Landau's excitation-based argument for superfluids---that at temperature
T=0 the normal fluid mass density ${\ensuremath{\rho}}_{N}$ is zero---should apply to supersolids. Together these two arguments imply that at low
T there is a missing mass density. We attribute this to a supersolid density ${\ensuremath{\rho}}_{L}$, with ${\ensuremath{\rho}}_{L}\ensuremath{\equiv}\ensuremath{\rho}\ensuremath{-}{\ensuremath{\rho}}_{S}\ensuremath{-}{\ensuremath{\rho}}_{N}$ and a momentum-bearing supersolid velocity
vLi. We derive the macroscopic dynamical equations for this system with three momentum-bearing macroscopic velocities
(vLi,vNi,vSi) using Onsager's irreversible thermodynamics. For slow spatial variations,
vSi accelerates along the negative gradient of the chemical potential or mass, like for an ordinary superfluid, and
vNi is subject to drag against
vLi with a characteristic drag time $\ensuremath{\tau}$. There are three contributions to the acceleration of
vLi: drag against
vNi, along the negative gradient of the chemical potential or mass, and due to the forces of lattice elasticity. For high-frequency motion relative to the drag time $\ensuremath{\tau}$, ${\ensuremath{\rho}}_{L}$ moves independently of ${\ensuremath{\rho}}_{N}$. The normal modes for such a system have a crossover frequency below which the normal fluid velocity
vNi is dragged into motion with
vLi (so the theory reduces to the
vSi,vNi theory of Andreev and Lifshitz) and above which the normal fluid velocity
vNi is an independent variable. For an isotropic lattice we study the transverse and longitudinal responses. A frequency-dependent rotational inertia experiment is predicted to show, at frequencies that are low relative to $\ensuremath{\tau}$, that ${\ensuremath{\rho}}_{L}$ and ${\ensuremath{\rho}}_{N}$ move together but that at high frequencies ${\ensuremath{\rho}}_{L}$ moves much more than ${\ensuremath{\rho}}_{N}$.
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.
We study the drag force on objects moving in a Fermi superfluid at velocities on the order of the Landau velocity
vL. The expectation has been that
vL is the critical velocity beyond which the drag force starts to increase toward its normal-state value. This expectation is challenged by a recent experiment measuring the heat generated by a uniformly moving wire immersed in superfluid
3He. We introduce the basis for the calculation of the drag force on a macroscopic object using the Fermi-liquid theory of superfluidity. As a technical tool in the calculations, we propose a boundary condition that describes diffuse reflection of quasiparticles from a surface on a scale that is larger than the superfluid coherence length. We calculate the drag force on steadily moving objects of different sizes. For an object that is small compared to the coherence length, we find a drag force that is in accordance with the expectation. For a macroscopic object, we need to take into account the spatially varying flow field around the object. At low velocities, this arises from ideal flow of the superfluid. At higher velocities, the flow field is modified by excitations that are created when the flow velocity locally exceeds
vL. The flow field causes Andreev reflection of quasiparticles and thus leads to change in the drag force. We calculate multiple limiting cases for a cylinder-shaped object. In the absence of quasiparticle-quasiparticle collisions, we find that the critical velocity is larger than
vL and the drag force (per cross-sectional area) at
2vL is reduced by an order of magnitude compared to the case of a small object. In a collision-dominated limit, the flow shows signs of instability at a velocity below
vL.
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.