Dynamics of superfluid 4He: Two-scale approach

Abstract

The paper explores the two scale approach to the incompressible dynamics of superfluid 4He. The fluid is described by a system of equations: Navier–Stokes and Euler equations for the macroscopic normal and superfluid velocity fields respectively. The two equations couple via a mutual friction force exerted on superfluid (quantum) vortices by the normal component. The magnitude of this force, calculated via the analysis of dynamics of quantum vortices in the microscopic scale, is proportional to the value of the counterflow (relative velocity of two helium components) and to the density of quantum vortices. The latter is determined by the generalized Vinen equation, adequate for flows having a net macroscopic vorticity. The generalized equation includes the drift of the anisotropic vortex tangle caused by Magnus force. The derived system of equations is applied first to the analysis of steady state solution of rotating turbulence, and then to the numerical analysis of formation of plane Couette flow between two infinite parallel material surfaces z = 0 and z = D . For t < 0 both surfaces and the fluid are at rest, then at t = 0 one material surface starts moving along the x axis with velocity V 0 . The viscosity forces cause the motion of normal component and the counterflow which make the line-length density grow, and the two components become coupled by the mutual friction. The fact that superfluid velocity tends to match with the normal velocity makes that ω s ≠ 0 , which implies polarization and drift of the tangle. At a given temperature the dynamics depends solely on D V 0 .