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

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 R (dimensionless, ratio of inertial to viscous forces in incompressible fluids). For example, the drag coefficient cD of objects with similar shapes moving in fluids is expressed by a universal function of R. Certain studies introduced similar dimensionless numbers, that is, the superfluid Reynolds number Rs, 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 cD and Rs 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.