# A pendulum in a superfluid

Imagine to submerge a pendulum in a supefluid. Of course we assume an ideal pendulum, whose joint does not freeze or deteriorate due to the extremely low temperature. We also assume the superfluid to be at zero temperature, so we can neglect its normal component.

What happens to its oscillations? Are they somehow damped? Or do they keep a constant amplitude?

In other words: can some energy be transferred from the pendulum to the superfluid (for example, thanks to the excitation of sound waves in the superfluid)?

If the pendulum is fully submerged (and the velocity of the sphere is sufficiently small) then the flow around the sphere is Stokes flow, and the drag is proportional to viscosity. As a result there is no drag in a zero temperature superfluid. If the sphere is only partially submerged, then it can excite surface waves which take away energy. This setup is not completely academic — torsion pendulums have been used to measure the viscosity of liquid helium.

Further remarks:

1. The drag is described by Stokes formula $$F=6\pi \eta RV$$, where $$\eta$$ is the viscosity of the normal fluid, $$R$$ is the radius of the sphere, and $$v$$ is the velocity relative to the fluid. Drag vanishes if $$\eta$$ vanishes, or the normal density vanishes.

2. If the sphere is only partially submerged then there is a free surface and Stokes solution does not apply. Indeed, more generally, d'Alembert's paradox does not apply. An object will generate a bow wave in an inviscid fluid, and this leads to drag.

3. Every superfluid has a critical velocity. If the velocity of the object exceeds that velocity, then superfluidity breaks down, even at very small temperature.

• Thanks. Can you please further comment on why if the sphere is only partially submerged, then it can excite surface waves? In addition: what happens, in the fully-submerged case, if the pendulum velocity is not sufficiently small? Oct 10, 2022 at 7:34
• Stokes flow? Stokes flow is the type of flow where viscosity is dominant and the inertial forces negligible. The almost potential flow in the superfluid component is the opposite of that. The normal viscous component may behave Stokes-like or not depending on the Reynolds number - as any other fluid. Oct 10, 2022 at 9:12
• But if you go to the low temperature limit and zero viscosity limit, than you should look at the potential flow, not the Stokes flow. Oct 10, 2022 at 9:18
• @VladimirFГероямслава The Stokes solution for flow around a sphere is potential flow. For zero viscosity there is no drag (this is sometimes called d'Alembert's paradox). For small $\eta$ we get Stokes famous result $F=6\pi\eta Rv$. Oct 10, 2022 at 14:16
• Even with zero viscosity, are there not compressive waves that dissipate energy from the pushing action of the pendulum? Oct 10, 2022 at 16:42