Hydromagnetic stability of current-induced flow in a small gap between concentric rotating cylinders

TLDR: In this paper, the stability analysis of the flow of a viscous electrically conducting fluid between concentric rotating cylinders in the presence of an axial magnetic field is extended to the case where the primary flow includes a pressure gradient acting in the azimuthal direction.

Abstract: The stability analysis of the flow of a viscous electrically conducting fluid between concentric rotating cylinders in the presence of an axial magnetic field is extended to the case where the primary flow includes a pressure gradient acting in the azimuthal direction. The pressure gradient is produced electromagnetically by the interaction of a superimposed radial current and the uniform axial magnetic field. The assumption of small gap approximation is made and the governing equations with respect to both axisymmetric and non–axisymmetric three–dimensional disturbances are derived and solved by a direct numerical procedure. A parametric study covering wide ranges of Q , the Hartmann number which represents the strength of axial magnetic field, and β, a parameter characterizing the ratio of current induced and rotation velocities, is conducted for weakly conducting cylinders and the situation of practical interest where the outer cylinder is stationary and the inner cylinder is rotating. The area where the onset mode is non–axisymmetric is shown in the plane β, Q ). It is found that the most stable state occurs approximately along a critical curve (β + 4.3) Q 2 + 56250(β + 3.75) = 0 and the critical axial wavenumber always has discontinuity when the parameters Q and β cross this curve. The critical mode transition of the onset of instability will be demonstrated in detail and results for the critical wavenumber and the critical Taylor number are presented. The corresponding values of the radial current density required for the appearance of secondary flow are also determined.