Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field

When a ferromagnetic fluid with a horizontal free surface is subjected to a uniform vertical applied magnetic field B0, it is known (Cowley & Rosensweig 1967) that the surface may be unstable when the field strength exceeds a certain critical value Bc. In this paper we consider, by means of an energy minimization principle, the possible forms that the surface may then take. Under the assumption that |μ − 1| [Lt ] 1 (where μ is the magnetic permeability of the fluid), it is shown that when B0 is near to Bc there are three equilibrium configurations for the surface: (i) flat surface, (ii) stationary hexagonal pattern, (iii) stationary square pattern. Configuration (i) is stable for B0 < Bc, (ii) is stable for B0 > Bc and B0−Bc sufficiently small, and (iii) is stable for some higher values of B0. In each configuration the fluid is static, and the surface is in equilibrium under the joint action of gravity, surface tension, and magnetic forces. The amplitude of the surface perturbation in cases (ii) and (iii) is calculated, and hysteresis effects associated with increase and decrease of B0 are discussed.