$$Re = \frac{\rho v L}{\mu}$$
where:
* $\rho$ is the density of the fluid
* $v$ is the velocity of the fluid
* $L$ is the characteristic length of the flow
* $\mu$ is the dynamic viscosity of the fluid
In superfluid helium, the dynamic viscosity is zero at temperatures below the lambda point, which is about 2.17 K. This means that superfluid helium flows without any friction, and Reynolds similitude is undefined.
However, it has been proposed that a quantum viscosity, which is a type of viscosity that arises from the quantum nature of the fluid, could exist in superfluid helium. If quantum viscosity does exist, then it would be possible to measure Reynolds similitude in superfluid helium by using a technique called torsional oscillator.
A torsional oscillator is a device that consists of a disk suspended from a wire. When the disk is twisted and released, it will oscillate back and forth. The frequency of the oscillations is determined by the moment of inertia of the disk and the torsional stiffness of the wire.
If a superfluid helium bath is placed around the torsional oscillator, the quantum viscosity of the helium will cause the disk to oscillate more slowly. The amount of damping depends on the quantum viscosity of the helium, and it can be used to measure Reynolds similitude.
Measuring Reynolds similitude in superfluid helium could help to demonstrate the existence of quantum viscosity. This would be a significant discovery, as it would provide new insights into the quantum nature of fluids.
