Homoclinic snaking near the surface instability of a polarizable fluid

TL;DR

This paper investigates localized hexagon patches on a magnetic fluid surface, explaining their stability and formation through energy principles and homoclinic snaking, with implications for pattern formation in physical systems.

Contribution

It introduces a Hamiltonian-based energy functional and numerical methods to analyze localized hexagon patches and their snaking behavior in ferrofluids.

Findings

Localized hexagon patches are stabilized near the Rosensweig instability.

Existence of Maxwell points explains the energy balance of patches.

Homoclinic snaking of patches matches experimental observations.

Abstract

We report on localized 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 neighborhood of the unstable branch of the domain covering hexagons of the Rosensweig instability upon which the patches equilibrate and stabilise. They are found to co-exist 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 provides 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 localized hexagons in the Young-Laplace equation coupled to the Maxwell equations. We find 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 2D localized patches of cellular patterns in many physical systems.