Mechanical momentum transfer in wall-bounded superfluid turbulence

In classical turbulence the kinematic viscosity $\nu$ is involved in two phenomena. The first is the energy dissipation and the second is the mechanical momentum flux toward the wall. In superfluid turbulence the mechanism of energy dissipation is different, and it is determined by an effective viscosity which was introduced by Vinen and is denoted as $\nu'$. In this paper we show that in superfluid turbulence the transfer of mechanical momentum to the wall is caused by the presence of a quantum vortex tangle, giving rise to another effective "momentum" viscosity that we will denote as $\nu_\text{m}(T)$. The temperature dependence of the second effective viscosity is markedly different from Vinen's effective viscosity $\nu'(T)$. We show that the notion of vortex-tension force, playing an important role in the theory of quantum turbulence, can be understood as the gradient of the Reynolds stress tensor which is in fact determined by the second newly defined kinematic viscosity $\nu_\text{m}(T)$.