  Analytic arguments have been advanced that the degree of differential
rotation in a neutron star depends on whether the topology of the internal
magnetic field is open or closed. To test this assertion, the
ideal-magnetohydrodynamics solver PLUTO is employed to investigate numerically
the flow of an incompressible, viscous fluid threaded by a magnetic field with
open and closed topologies in a conducting, differentially rotating, spherical
shell. Rigid body corotation with the outer sphere is enforced on the Alfv\'en
time-scale, along magnetic field lines that connect the northern and southern
hemispheres of the outer sphere. Along other field lines, however, the behavior
is more complicated. For example, an initial point dipole field evolves to
produce an approximately closed equatorial flux tube containing at least one
predominantly toroidal and approximately closed field line surrounded by a
bundle of predominantly toroidal but open field lines. Inside the equatorial
flux tube, the field-line-averaged magnetic tension approaches zero, and the
fluid rotates differentially, adjusting its angular velocity on the viscous
time-scale to match the boundary conditions on the flux tube's toroidal
surface. Outside the equatorial flux tube the differential rotation increases,
as the magnetic tension averaged along open field lines decreases.
