THE SPIN-UP PROBLEM IN HELIUM-II

The laminar spin-up of Helium II is studied by solving the linearized equations of motion for the normal and superfluid components and the q unatized vortex lines in a simple case. The fluid is taken to be confi ned between tow parallel planes whose angular velocity increases at a small, steady rate. The vortex lines are treated as a continuum. No di rect interactions between the vortex lines and the walls are included. Two mechanisms are identified for the transfer of angular momentum fr om the container to the interior fluid. In the first place, classical Ekman pumping occurs in the normal fluid component. Secondly, mutual f riction between the normal Ekman layer and the vortex lines produces a n (Ekman-like) secondary flow in the superfluid component. In both mec hanisms, mutual friction in the interior couples the normal and superf luid components together, so that both components spin up. Normal-flui d Ekman pumping is found to dominate at temperatures close to the lamb da-point (T(lambda) = 2.17 K), while the second mechanism becomes prog ressively more important at lower temperatures. In the small-Ekman-num ber limit, when the vertical container dimension 2a is much larger tha n the Ekman layer thickness, the spin-up time (i.e., the time lag betw een the container and the interior fluid) for both components is t(spi n-up) almost-equal-to f(T) aOMEGA0(-1/2), where OMEGA0 is the angular velocity and f(T) is a decreasing function of temperature. Although so me experimental spin-up times in He II have been reported in the liter ature, their analysis involves many uncertainties. Thus, new experimen ts to test this model should be highly desirable.