Spin-up of a two-component superfluid: self-consistent container feedback

The coupled dynamic response of a rigid container filled with a two-component superfluid undergoing Ekman pumping is calculated self-consistently. The container responds to the back-reaction torque exerted by the viscous component of the superfluid and an arbitrary external torque. The resulting motion is described by a pair of coupled integral equations for which solutions are easily obtained numerically. If the container is initially accelerated impulsively then set free, it relaxes quasi-exponentially to a steady state over multiple time scales, which are a complex combination of the Ekman number, superfluid mutual friction coefficients, the superfluid density fraction, and the varying hydrodynamic torque at different latitudes. The spin-down of containers with relatively small moments of inertia (compared with that of the contained fluid) depends weakly on the above parameters and occurs faster than the Ekman time. When the fluid components are initially differentially rotating, the container can 'overshoot' its asymptotic value before increasing again. When a constant external torque is applied, the superfluid components rotate differentially and non-uniformly in the long term. For an oscillating external torque, the amplitude and phase of the oscillation are most sensitive to the driving frequency for containers with relatively small moments of inertia. Applications to superfluid helium experiments and neutron stars are also discussed.