The nonaxisymmetric 'kink-type' Tayler instability (TI) of toroidal magnetic
fields is studied for conducting incompressible fluids of uniform density
between two infinitely long cylinders rotating around the same axis. It is
shown that for resting cylinders the critical Hartmann number for the unstable
modes does not depend on Pm. By rigid rotation the instability is suppressed
where the critical ratio of the rotation velocity and the Alfven velocity of
the field (only) slightly depends on the magnetic Prandtl number Pm. For Pm=1
the rotational quenching of TI takes its maximum. Rotation laws with negative
shear (i.e. d\Omega/dR<0) strongly destabilize the toroidal field if the
rotation is not too fast. For sufficiently high Reynolds numbers of rotation
the suppression of the nonaxisymmetric magnetic instability always dominates.
The angular momentum transport of the instability is anticorrelated with the
shear so that an eddy viscosity can be defined which proves to be positive. For
negative shear the Maxwell stress of the perturbations remarkably contributes
to the angular momentum transport. We have also shown the possibility of
laboratory TI experiments with a wide-gap container filled with fluid metals
like sodium or gallium. Even the effect of the rotational stabilization can be
reproduced in the laboratory with electric currents of only a few kAmp.Comment: 9 pages, 11 figures, sub