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

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Abstract

<jats:p>When a ferromagnetic fluid with a horizontal free surface is subjected to a uniform vertical applied magnetic field <jats:italic>B</jats:italic><jats:sub>0</jats:sub>, it is known (Cowley & Rosensweig 1967) that the surface may be unstable when the field strength exceeds a certain critical value <jats:italic>B</jats:italic><jats:sub><jats:italic>c</jats:italic></jats:sub>. 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 <jats:italic>B</jats:italic><jats:sub>0</jats:sub> is near to <jats:italic>B</jats:italic><jats:sub><jats:italic>c</jats:italic></jats:sub> there are three equilibrium configurations for the surface: (i) flat surface, (ii) stationary hexagonal pattern, (iii) stationary square pattern. Configuration (i) is stable for <jats:italic>B</jats:italic><jats:sub>0</jats:sub> < <jats:italic>B</jats:italic><jats:sub><jats:italic>c</jats:italic></jats:sub>, (ii) is stable for <jats:italic>B</jats:italic><jats:sub>0</jats:sub> > <jats:italic>B</jats:italic><jats:sub><jats:italic>c</jats:italic></jats:sub> and <jats:italic>B</jats:italic><jats:sub>0</jats:sub>−<jats:italic>B</jats:italic><jats:sub><jats:italic>c</jats:italic></jats:sub> sufficiently small, and (iii) is stable for some higher values of <jats:italic>B</jats:italic><jats:sub>0</jats:sub>. 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 <jats:italic>B</jats:italic><jats:sub>0</jats:sub> are discussed.</jats:p>

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