Linear and non-linear stability of helical flow of a hetero-geneous conducting fluid

Abstract

The problem of linear and nonlinear stability of a helical flow of a perfectly conducting heterogeneous fluid between two coaxial cylinders in the presence of an azimuthal magnetic field and a radial gravitational force is discussed. In the case of linear stability the problem has been formulated by the normal mode method and the analysis has been carried out by reducing the perturbation equations to a Sturm-Liouville system. It is found that a necessary condition for instability is that the algebraic sum of hydrodynamic, magnetohydrodynamic and swirling Richardson numbers must be less than one quarter somewhere in the fluid. A semi-circle theorem similar to that of Howard is also obtained. In particular it is found that when gravitational force balances the centrifugal force of the swirling motion, the heterogeneous conducting fluid behaves as if it is homogeneous as far as the condition for stability is concerned. In the case of nonlinear stability the problem has been formulated by the energy method and a universal stability estimate, namely a stability limit for motions subject to arbitrary nonlinear disturbances is obtained in terms of Alfven number and Richardson number, J, for the flow. In the case of hydrodynamic flow by letting magnetic field tend to zero, it is found that the motion is stable if J≽o.