  We present numerical simulations of the flow in a rapidly rotating
cylindrical annulus. We show that at the rotation rates relevant to the
magneto-rotational instability, the flow is strongly constrained by the
Taylor-Proudman theorem. As a result, it is controlled almost entirely by the
end-plates. We then consider two possible options for minimizing these
end-effects, namely (i) simply taking a very long cylinder, and (ii) splitting
the end-plates into a series of differentially rotating rings. Regarding option
(i), we show that the cylinder would have to be hundreds of times as long as it
is wide before end-effects become unimportant in the interior. Since this is
clearly not feasible, we turn to option (ii), and show that in order to obtain
a smooth angular velocity profile, the end-plates would have to be split into
around ten rings. If the end-plates are split into fewer rings, perhaps 3-5,
the angular velocity profile will not be smooth, but will instead consist of a
series of Stewartson layers at the boundaries from one ring to the next. We
suggest therefore that the instabilities one obtains in this system will be the
familiar Kelvin-Helmholtz instabilities of these Stewartson layers, rather than
the magneto-rotational instability. At best, one might hope to obtain the MRI
superimposed on these Kelvin-Helmholtz modes. Any subsequent interpretation of
results is thus likely to be quite complicated.
