The stability of axisymmetric Taylor-Couette flow in hydromagnetics

The linear marginal instability of an axisymmetric MHD Taylor-Couette flow of infinite vertical extension is considered. The dependence of the flow stability on magnetic Prandtl number, Pm, and gap-width between rotating cylinders is investigated. There is an unstability for flows with resting outer cylinder even without a magnetic field. Nevertheless, there are solutions with smaller critical Reynolds numbers for certain (weak) magnetic fields. These solutions, however, exist for not too small Pm or large enough gap. For hydrodynamically stable flow, we found that a magnetic field amplitude always exsits where the critical Reynolds number is minimal. In all calculated cases the minimal critical Reynolds numbers are running with 1/Pm for small Pm and the critical Reynolds numbers exceed value of 10^6 for Pm of sodium (10^{-5}) or gallium (10^{-6}).