Hydromagnetic Stability of a Conducting Fluid in a Circular Magnetic Field

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Abstract
The theory of viscous flow between two rotating coaxial cylinders as developed by Taylor and Chandrasekhar is extended to the case when the fluid is an electrical conductor and a circular magnetic field (i.e., one whose lines of force are concentric with the cylinder walls) is present. The equations governing marginal stability are derived, and boundary conditions for perfectly conducting cylinders (Fermi boundary conditions) are formulated for two cases when the difference in cylinder radii is small compared to their mean. In the first case, co‐rotating cylinders, the underlying characteristic value problem is solved by a variational method developed by Chandrasekhar to show that convective instability rather than oscillatory overstability will occur for realizable magnetic field strengths. In the second case, co‐rotating and counter‐rotating cylinders, the underlying characteristic value problem is solved by an expansion in orthogonal functions method developed by Chandrasekhar to determine critical Taylor numbers for marginal stability. The magnetic field inhibits the onset of instability, but this effect is quite small as the hydromagnetic interaction involves displacement but not distortion of the magnetic lines of force.