Superfluid spherical Couette flow

Abstract

We solve numerically for the first time the two-fluid, Hall--Vinen--Bekarevich--Khalatnikov (HVBK) equations for a He-II-like superfluid contained in a differentially rotating, spherical shell, generalizing previous simulations of viscous spherical Couette flow (SCF) and superfluid Taylor--Couette flow. In axisymmetric superfluid SCF, the number of meridional circulation cells multiplies as $\Rey$ increases, and their shapes become more complex, especially in the superfluid component, with multiple secondary cells arising for $\Rey > 10^3$. The torque exerted by the normal component is approximately three times greater in a superfluid with anisotropic Hall--Vinen (HV) mutual friction than in a classical viscous fluid or a superfluid with isotropic Gorter-Mellink (GM) mutual friction. HV mutual friction also tends to "pinch" meridional circulation cells more than GM mutual friction. The boundary condition on the superfluid component, whether no slip or perfect slip, does not affect the large-scale structure of the flow appreciably, but it does alter the cores of the circulation cells, especially at lower $\Rey$. As $\Rey$ increases, and after initial transients die away, the mutual friction force dominates the vortex tension, and the streamlines of the superfluid and normal fluid components increasingly resemble each other. In nonaxisymmetric superfluid SCF, three-dimensional vortex structures are classified according to topological invariants.