  The nonaxisymmetric azimuthal magnetorotational instability is studied for
hydromagnetic Taylor-Couette flows between cylinders of finite electrical
conductivity. We find that the magnetic Prandtl number Pm determines whether
perfectly conducting or insulating boundary conditions lead to lower Hartmann
numbers for the onset of instability. Regardless of the imposed rotation
profile, for small Pm the solutions for perfectly conducting cylinders become
unstable for weaker magnetic fields than the solutions for insulating
cylinders. The critical Hartmann and Reynolds numbers form monotonic functions
of the ratio of the electrical conductivities of the cylinders and the fluid,
such that a ratio of about 10 provides a very good approximation to perfectly
conducting cylinders, and a ratio of about 0.1 a very good approximation to
insulating cylinders. These results are of particular relevance for the
super-rotating case where the outer cylinder rotates faster than the inner one;
in this case the critical onset values are substantially different for
perfectly conducting versus insulating boundary conditions. An experimental
realization of the super-rotating instability, with liquid sodium as the fluid
and cylinders made of copper, would require an electric current of at least
33.5 kAmp running along the central axis.
