Shalybkov Dima
A.F. Ioffe Institute for Physics and Technology, 194021 St. Petersburg, Russia.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jan;73(1 Pt 2):016302. doi: 10.1103/PhysRevE.73.016302. Epub 2006 Jan 19.
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.
研究了具有方位磁场的耗散泰勒-库埃特流的线性稳定性。与理想流不同，磁场仅是半径的固定函数，仅具有两个参数：内筒半径与外筒半径之比η，以及外筒和内筒上磁场值之比μB。0<μB<1/η的磁场使流动稳定，称为稳定磁场。无电流磁场（μB = η）是稳定磁场。不稳定磁场的值（或哈特曼数）超过某个临界值时，会使包括无磁场时稳定的流动在内的所有流动失稳。即使没有旋转，这种不稳定性仍然存在。对于无磁场时稳定的流动，不稳定模式位于轴向波数的某个区间内。对于临界哈特曼数，该区间长度为零，并随哈特曼数的增加而增大。对于每种旋转定律，临界哈特曼数和不稳定轴向波数区间的长度都是相同的。仅对于m = 0的腊肠模和m = 1的扭结模存在临界哈特曼数。腊肠模是接近Ha = 0点时最不稳定的模式，扭结模是接近临界哈特曼数时最不稳定的模式。从腊肠不稳定性到扭结不稳定性的转变取决于普朗特数Pm，对于Pm = 1，这种转变发生在临界哈特曼数的大约一半附近，对于Pm = 10^(-5)，则发生在接近临界哈特曼数处。扭结模的临界哈特曼数较小。对于m = 0模式，流动稳定性不依赖于磁普朗特数。对于m = 0和m = 1模式的临界哈特曼数也是如此。使液态金属泰勒-库埃特流失稳的磁场典型值约为10^2 G。
