Effect of Joule dissipation on the convective instability of a conducting liquid with flow in a magnetic field

Abstract

The results of [1] are extended to the case when the Joule dissipation leads to a nonlinear profile of the unperturbed temperature of the liquid. Convective instability of a conducting liquid, with flow in a magnetic field directed perpendicular to the flow, with a temperature-dependent distribution of the conductivity which is nonhomogeneous in the direction of action of the electromagnetic force, was discussed in [1], neglecting Joule dissipation. This type of approach permitted investigating an energy equation without electromagnetic terms, which to a certain degree facilitated the solution of the problem. In many cases, however, the Joule dissipation is considerable and may exert a considerable effect on the development of convective instability. Thus, without taking account of Joule evolution of heat, instability can arise only with positive values of the Rayleigh number, exceeding some critical value, while, at the same time, Joule dissipation may lead to a situation in which instability will develop also with negative values of the Rayleigh number, i.e., under conditions when the state without the evolution of Joule heat is absolutely stable.