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The theory of stability of the flow of a viscous, electrically conducting fluid between rotating cylinders in the presence of an axial magnetic field is extended to the case where the cylinders are permeable and the primary flow includes a radial component. Numerical results pertaining to the stationary axially symmetric modes are presented, and the asymptotic stability behaviour for large values of the radial Reynolds number is derived.

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SynopsisA small perturbation analysis is carried out to determine the stability of a fluid containing two layers of dBusing solutes in a common solvent and acted upon by a uniform gravitational field. It is found that instability can arise even though the unperturbed diffusion does not lead to the formation of a density inversion within the fluid.

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In considering the onset of instability of an electrically conducting fluid between rotating permeable perfectly conducting cylinders in an applied axial magnetic field, it is found that oscillatory axially symmetric modes occur at large values of Hartmann number, in addition to the usual stationary modes. Results are presented showing the effect of the oscillatory modes on the criterion of onset of instability.The asymptotic behaviour of the stability criterion is considered in the limit of very large radial Reynolds number, and also in the limit where both the radial Reynolds number and the Hartmann number are large.

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The transition of the onset of instability from stationary modes to oscillatory modes for an incompressible, conducting Couette flow between two coaxial, perfectly conducting, non-permeable, rotating cylinders under the influence of an axially applied magnetic field is considered. Results for three cases are reported. These pertain to flow between (1) a rotating inner wall with a stationary outer wall, (2) counterrotating walls, and (3) corotating walls. It is found that, for high values of the Hartmann number, there may exist some range of convective wavenumbers for which neither of the two lowest modes of axisymmetrical disturbances will become stationary. Within this range, the neutral stability curve is determined by a complex-conjugate pair of oscillatory axisymmetrical modes of equal stability. The oscillatory modes may, in fact, become more critical than the stationary modes. It is demonstrated that the approximation of replacing the angular speed by its average value, combined with the assumption of a narrow gap between the cylindrical walls, eliminates the oscillatory axisymmetrical modes.