On stationary convection and oscillatory motions in ferromagnetic convection in a ferrofluid layer

It is shown analytically that the 'principle of the exchange of stabilities' (PES), in general, is not valid in ferromagnetic convection in a ferrofluid layer, for the case of free boundaries and hence a sufficient condition is derived for the validity of the PES. Upper bounds for the complex growth rate are then obtained. It is proved that the complex growth rate sigma = sigma(r) + i sigma(i) (where sigma(r) and sigma(i) are, respectively, the real and imaginary parts of sigma) of an arbitrary oscillatory motion of growing amplitude, in ferromagnetic convection in a ferrofluid layer, for the case of free boundaries lies inside a semicircle in the right half of the sigma(r)sigma(i)-plane whose center is at the origin and (radius)(2) = RM1/P-r, where R is the Rayleigh number,M-1 is the magnetic number and P-r is the Prandtl number. Further, bounds for the case of rigid ferromagnetic boundaries are also derived separately. (C) 2011 Elsevier B.V. All rights reserved.