Pinch instabilities in Taylor-Couette flow

Abstract

The linear stability of the dissipative Taylor-Couette flow with an azimuthal magnetic field is considered. Unlike ideal flows, the magnetic field is a fixed function of a radius with two parameters only: a ratio of inner to outer cylinder radii, eta, and a ratio of the magnetic field values on outer and inner cylinders, muB. The magnetic field with 0<muB<1/eta stabilizes the flow and is called a stable magnetic field. The current free magnetic field (muB=eta) is the stable magnetic field. The unstable magnetic field, which value (or Hartmann number) exceeds some critical value, destabilizes every flow including flows which are stable without the magnetic field. This instability survives even without rotation. The unstable modes are located into some interval of the axial wave numbers for the flow stable without magnetic field. The interval length is zero for a critical Hartmann number and increases with an increasing Hartmann number. The critical Hartmann numbers and length of the unstable axial wave number intervals are the same for every rotation law. There are the critical Hartmann numbers for m=0 sausage and m=1 kink modes only. The sausage mode is the most unstable mode close to Ha=0 point and the kink mode is the most unstable mode close to the critical Hartmann number. The transition from the sausage instability to the kink instability depends on the Prandtl number Pm and this happens close to one-half of the critical Hartmann number for Pm=1 and close to the critical Hartmann number for Pm=10(-5). The critical Hartmann numbers are smaller for kink modes. The flow stability does not depend on magnetic Prandtl numbers for m=0 mode. The same is true for critical Hartmann numbers for both m=0 and m=1 modes. The typical value of the magnetic field destabilizing the liquid metal Taylor-Couette flow is approximately 10(2) G.