Quantum viscosity and the Reynolds similitude of a pure superfluid

The 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 $\mathcal{R}$ (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 $\mathcal{R}$. Certain studies introduced similar dimensionless numbers, that is, the superfluid Reynolds number ${\mathcal{R}}_s$, to characterize turbulent flows in superfluids. However, the applicability of the similitude to inviscid quantum fluids is nontrivial as the original theory is applicable to viscous fluids. This Letter proposes a method to verify the similitude using current experimental techniques in quantum liquid He II. A highly precise relation between $c_D$ and ${\mathcal{R}}_s$ 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. The Reynolds similitude in superfluids proves the quantum viscosity of a pure superfluid and can facilitate a unified mutual development of classical and quantum hydrodynamics; the concept of quantum viscosity provides a practical correspondence between classical and quantum turbulence as a dissipative phenomenon.

Figure 1

Schematics of the wake behind a sphere moving from the right to left in quantum liquid He II. (a) Drag is zero without quantum vortices at $T=0$. (b) Coarse-grained quantum vortices reproduce a turbulent wake with high $\mathcal{R}(={\mathcal{R}}_s)$ according to the Reynolds similitude extended to superfluids. The inset represents a microscopic view of quantum vortices in the turbulent wake. (c) At finite temperatures in the hydrodynamic regime ($l_{\mathrm{MFP}}\ll d$), quasiparticles form a Stokes flow in the presence of a turbulent wake in the superfluid component. (d) Quasiparticles form a dilute gas and the drag is caused by their ballistic scattering in the Knudsen regime ($l_{\mathrm{MFP}}\gg d$).

Figure 2

Plots of the relation of ${\mathcal{R}}_s$ to the diameter $d$ (solid) and the terminal speed ${\overline{u}}_T$ (dashed) for spheres in He II under vapor pressure at $T=0$ K. Blue, yellow, and green curves correspond to the results for $\gamma =1.11$ (${}^4\mathrm{He}$ crystal), $\gamma =4$, and $\gamma =45.8$ (iron), respectively.

Figure 3

Temperature dependence of (left) ${\mathcal{R}}_n/{\mathcal{R}}_s$, $l_{\mathrm{ph}}$, and (right) ${\eta }_n^{\prime }$ with $C=3/5$ for spheres of radius $d={10}^{-4}$, ${10}^{-3}$, and ${10}^{-2}\mathrm{m}$. For reference, we also plot ${\eta }_{n,\mathrm{ph},\mathrm{ro},s}$ and ${\eta }_n^{\prime }$. We have ${\eta }_n^{\prime }\propto T^4$ in the Knudsen limit ($l_{\mathrm{ph}}\gg d$).

Figure 4

Plot of thermal correction ${\delta }_{\mathrm{th}}$ for $d={10}^{-3}$ m. The solid curves shows the contour of ${\delta }_{\mathrm{th}}=0.1$ for $d={10}^{-4}$, ${10}^{-3}$, and ${10}^{-2}$ m.