A Simple Phenomenological Model for the Effective Kinematic Viscosity of Helium Superfluids

A number of experiments where quantum turbulence in helium superfluids has been generated by various means (such as towed/oscillating grids, thermal counterflow, pure superflow, spin-down, ion/vortex rings emission) displays a temporal decay of the observed vortex line density, of the power law form L=Γt−3/2 at late times. The prefactor, Γ, in analogy with classical homogeneous isotropic turbulence, allows deducing the temperature dependent effective kinematic viscosity, νeff, for turbulent helium superfluids. It appears to be a robust quantity, independent of methods of generating quantum turbulence and detecting the decaying vortex line density. We present a simple phenomenological model to estimate νeff based on comparison of dissipation terms in equations of motion for classical viscous flow and vortex flow of a superfluid in a stationary normal fluid. This model leads to νeff≈κq, where q=α/(1−α′); α and α′ being dimensionless mutual friction parameters. Within the temperature range where mutual friction dissipation mechanism is dominant this simple model prediction agrees well with the experimental data and with the recent theoretical estimate of Roche, Barenghi and Leveque (Europhys. Lett. 87:54006, 2009).