The 2-3-4 spike competition in the Rosensweig instability

The horizontal free surface of a magnetic liquid (ferrofluid) pool turns unstable when the strength of a vertically applied uniform magnetic field exceeds a threshold. The instability, known as normal field instability or Rosensweig’s instability, is accompanied by the formation of liquid spikes either few, in small diameter pools, or many, in large diameter pools; in the latter case, the spikes are arranged in hexagonal or square patterns. In small pools where only few spikes – 2, 3 or 4 in this work – can be accommodated, their appearance/disappearance/re-appearance observed in experiments, as applied field strength varies, is investigated by computer-aided bifurcation and linear stability analysis. The equations of three-dimensional capillary magneto-hydrostatics give rise to a three-dimensional free boundary problem which is discretized by the Galerkin/finite element method and solved for multi-spike surface deformation coupled with magnetic field distribution simultaneously with a compact numerical scheme based on Newton iteration. Standard eigenvalue problems are solved in the course of parameter continuation to reveal the multiplicity and the stability of the emerging deformations. The computational predictions reveal selection mechanisms among equilibrium states and explain the corresponding experimental observations and measurements.

Small-amplitude static periodic patterns at a fluid–ferrofluid interface

We establish the existence of static doubly periodic patterns (in particular rolls, squares and hexagons) on the free surface of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetization law. A novel formulation of the ferrohydrostatic equations in terms of Dirichlet–Neumann operators for nonlinear elliptic boundary-value problems is presented. We demonstrate the analyticity of these operators in suitable function spaces and solve the ferrohydrostatic problem using an analytic version of Crandall–Rabinowitz local bifurcation theory. Criteria are derived for the bifurcations to be sub-, super- or transcritical with respect to a dimensionless physical parameter.

Dynamic Interfacial Phenomena at Water-Magnetic Fluid System Subject to Alternating Magnetic Field

Responses of a magnetic fluid interface adsorbed on a small permanent magnet in water container subjected to an alternating magnetic field were studied with a high-speed video camera system. The directions of the external alternating magnetic field were parallel and anti-parallel to that of the permanent magnet. It was found that the interface of water-magnetic fluid responds to the external alternating magnetic field in elongation and contraction with Rosensweig instability at the interface. Frequency characteristics of the interface response of water-magnetic fluid system subjected to alternating magnetic field were revealed over a wide frequency band experimentally.

Homoclinic snaking near the surface instability of a polarisable fluid

We report on localised patches of cellular hexagons observed on the surface of a magnetic fluid in a vertical magnetic field. These patches are spontaneously generated by jumping into the neighbourhood of the unstable branch of the domain-covering hexagons of the Rosensweig instability upon which the patches equilibrate and stabilise. They are found to coexist in intervals of the applied magnetic field strength parameter around this branch. We formulate a general energy functional for the system and a corresponding Hamiltonian that provide a pattern selection principle allowing us to compute Maxwell points (where the energy of a single hexagon cell lies in the same Hamiltonian level set as the flat state) for general magnetic permeabilities. Using numerical continuation techniques, we investigate the existence of localised hexagons in the Young–Laplace equation coupled to the Maxwell equations. We find that cellular hexagons possess a Maxwell point, providing an energetic explanation for the multitude of measured hexagon patches. Furthermore, it is found that planar hexagon fronts and hexagon patches undergo homoclinic snaking, corroborating the experimentally detected intervals. Besides making a contribution to the specific area of ferrofluids, our work paves the ground for a deeper understanding of homoclinic snaking of two-dimensional localised patches of cellular patterns in many physical systems.