The spin-up problem in Helium II

Abstract

The laminar spin-up of Helium II is studied by solving the linearized equations of motion for the normal and superfluid components and the quantized vortex lines in a simple case. The fluid is taken to be confined between two parallel planes whose angular velocity increases at a small, steady rate. The vortex lines are treated as a continuum. No direct interactions between the vortex lines and the walls are included. Two mechanisms are identified for the transfer of angular momentum from the container to the interior fluid. In the first place, classical Ekman pumping occurs in the normal fluid component. Secondly, mutual friction between the normal Ekman layer and the vortex lines produces an (Ekman-like) secondary flow in the superfluid component. In both mechanisms, mutual friction in the interior couples the normal and superfluid components together, so that both components spin up. Normal-fluid Ekman pumping is found to dominate at temperatures close to the λ-point (T_λ=2.17 K), while the second mechanism becomes progressively more important at lower temperatures. In the small-Ekman-number limit, when the vertical container dimension 2a is much larger than the Ekman layer thickness, the spin-up time (i.e., the time lag between the container and the interior fluid) for both components ist _spin-up≈f(T)aΩ ^−1/2₀ , where Ω₀ is the angular velocity andf(T) is a decreasing function of temperature. Although some experimental spin-up times in He II have been reported in the literature, their analysis involves many uncertainties. Thus, new experiments to test this model should be highly desirable.