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

Abstract

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.

Via hexagons to squares in ferrofluids: experiments on hysteretic surface transformations under variation of the normal magnetic field

“…At a first glance, the amplitude seems to be continuous in spite of the picture predicted by Gailitis (1977) and Friedrichs & Engel (2001). Obviously the height of Bifurcation parameter ε Figure 6.…”

“…It was uncovered by Cowley & Rosensweig (1967) soon after the synthesis of the first ferrofluids and thus has served as a "coat of arms" for the field of magnetic fluid research. The fascination stems in part from the fact that liquid crests which persist without motion are not a familiar experience (see In contrast to pattern formation in dissipative systems (Cross & Hohenberg 1993) the phenomenon can be described by an energy functional, which comprises hydrostatic, magnetic and surface energy (Gailitis 1969, Kuznetsov & Spektor 1976, Gailitis 1977. As the surface profile deviates from the flat reference state, the contributions of the hydrostatic and the surface energy grow whereas the magnetic energy decreases.…”

Mag(net)ic Liquid Mountains

“…e three terms make up a free-energy functional. Its minimization predicts a hexagonal array of spikes to be the first stable pattern, which appears due to a transcritical bifurcation [6]. We have checked the predicted scaling [6,7] by means of radioscopy.…”

“…Its minimization predicts a hexagonal array of spikes to be the first stable pattern, which appears due to a transcritical bifurcation [6]. We have checked the predicted scaling [6,7] by means of radioscopy. e green and blue data points in Fig.4 mark the hysteretic evolution of the pattern amplitude for increasing and decreasing induction, respectively [8].…”

“…e green and blue data points in Fig.4 mark the hysteretic evolution of the pattern amplitude for increasing and decreasing induction, respectively [8]. e solid lines give a fit to the solution of the amplitude equations [6]. Solving the Young-Laplace and Maxwell equations the full topography for the actual nonlinear magnetization has been calculated [9].…”