Convective instability of a ferrofluid

We clarify the conditions under which instability arises in the equilibrium of a nonuniformly heated ferrofluid in a gravitational field and a nonuniform magnetic field. The latter is, in the first place, responsible for the Archimedean buoyant forces, and in the second, gradients in the magnetic intensity result in the appearance of internal heat sources (magnetocaloric effect). As a rule, this effect is extremely weak and to take correct account of it requires that at the same time the compressibility of the fluid be taken into account in the equation of heat conduction. We show that it is precisely the neglect of compressibility that explains the erroneous conclusion, contradictory to the laws of thermodynamics, concerning the convective instability of an isothermal ferrofluid that was arrived at in a series of papers by B. M. Berkovskii. We formulate a dimensionless criterion that characterizes the stability of the equilibrium of a ferrofluid. In limiting cases of large or small cavities this criterion passes over to the ferrohydrodynamic analog of the usual Schwartzschild or Rayleigh criteria.