The quantization of vortex lines in superfluids requires the introduction of their density $C L(B r,t)$ in the description of quantum turbulence. The space homogeneous balance equation for $C L(t)$, proposed by Vinen on the basis of dimensional and p
hysical considerations, allows a number of competing forms for the production term $C P$. Attempts to choose the correct one on the basis of time-dependent homogeneous experiments ended inconclusively. To overcome this difficulty we announce here an approach that employs an inhomogeneous channel flow which is excellently suitable to distinguish the implications of the various possible forms of the desired equation. We demonstrate that the originally selected form which was extensively used in the literature is in strong contradiction with our data. We therefore present a new inhomogeneous equation for $C L(B r,t)$ that is in agreement with our data and propose that it should be considered for further studies of superfluid turbulence.
The relationship between the apparently unrelated physical quantities -- kinematic viscosity of liquid He-4, $ u$, and quantum of circulation, $kappa=2pi hbar/m_4$, where $hbar$ is the Planck constant and $m_4$ denotes the mass of the $^4$He atom --
is examined in the vicinity of the superfluid transition occurring due to Bose-Einstein condensation. A model is developed, leading to the surprisingly simple relation $ u approx kappa/6$. We critically examine the available experimental data for $^4$He relevant to this simple relation and predict the kinematic viscosity for the stretched liquid $^4$He along the $lambda$-line at negative pressures.
