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The linear stability of the dissipative Taylor-Couette flow with imposed azimuthal magnetic field is considered. Unlike to ideal flow, the magnetic field is fixed function of radius with two parameters only: a ratio of inner to outer cylinder radii and a ratio of the magnetic field values on outer and inner cylinders. The magnetic field with boundary values ratio greater than zero and smaller than inverse radii ratio always stabilizes the flow and called stable magnetic field below. The current free magnetic field is the stable magnetic field. The unstable magnetic field destabilizes every flow if the magnetic field (or Hartmann number) exceeds some critical value. This instability survives even without rotation (for zero Reynolds number). For the stable without the magnetic field flow, the unstable modes are located into some interval of the vertical wave numbers. The interval length is zero for critical Hartmann number and increases with increasing Hartmann number. The critical Hartmann numbers and the length of the unstable vertical wave numbers interval is the same for every rotation law. There are the critical Hartmann numbers for m = 0 sausage and m = 1 kink modes only. The critical Hartmann numbers are smaller for kink mode and this mode is the most unstable mode like to the pinch instability case. The flow stability do not depend on the magnetic Prandtl number for m = 0 mode. The same is true for critical Hartmann numbers for m = 0 and m = 1 modes. The typical value of the magnetic field destabilizing the liquid metal Taylor-Couette flow is order of 100 Gauss.

Taylor-Couette flow stability with toroidal magnetic field