Quantum viscosity and the Reynolds similitude in quantum liquid He-II

TL;DR

This paper investigates whether Reynolds similitude, a principle from classical hydrodynamics, applies to quantum liquid helium-II by establishing a precise relation between drag coefficient and a superfluid Reynolds number across different regimes.

Contribution

The study proposes a method to verify Reynolds similitude in quantum fluids and derives a precise relation between drag coefficient and superfluid Reynolds number in He-II.

Findings

Established a relation between drag coefficient and superfluid Reynolds number.

Verified applicability of Reynolds similitude in quantum liquid He-II.

Unified classical and quantum hydrodynamics through similitude.

Abstract

Reynolds similitude, a key concept in hydrodynamics, states that two phenomena of different length scales with a similar geometry are physically identical. Flow properties are universally determined in a unified way in terms of the Reynolds number ${\cal R}$ (dimensionless, ratio of inertial to viscous forces in incompressible fluids). For example, the drag coefficient $c_D$ of objects with similar shapes moving in fluids is expressed by a universal function of ${\cal R}$. Certain studies introduced similar dimensionless numbers, that is, the superfluid Reynolds number ${\cal R}_s$, to characterize turbulent flows in superfluids. However, the applicablity of the similitude to inviscid quantum fluids is nontrivial as the original theory is applicable to viscous fluids. This study proposed a method to verify the similitude using current experimental techniques in quantum liquid He-II. A highly precise relation between $c_D$ and ${\cal R}_s$ was obtained in terms of the terminal speed of a macroscopic body falling in He-II at finite temperatures across the Knudsen (ballistic) and hydrodynamic regimes of thermal excitations. Reynolds similitude in superfluids can facilitate unified mutual development of classical and quantum hydrodynamics.