Pattern formation on the free surface of a ferrofluid: spatial dynamics and homoclinic bifurcation

TL;DR

This paper demonstrates the existence of localized free surface patterns in ferrofluids near the instability threshold, using a Hamiltonian framework and bifurcation analysis to identify homoclinic solutions.

Contribution

It introduces a novel variational and Hamiltonian approach to analyze localized surface patterns in ferrofluids near the Rosensweig instability.

Findings

Existence of spatially localized free surface solutions.

Application of normal-form theory to identify homoclinic bifurcations.

Reduction of the problem to a finite-dimensional Hamiltonian system.

Abstract

We establish the existence of spatially localised one-dimensional free surfaces of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation law. It is shown that the ferrohydrostatic equations can be derived from a variational principle that allows one to formulate them as an (infinite-dimensional) spatial Hamiltonian system in which the unbounded free-surface direction plays the role of time. A centre-manifold reduction technique converts the problem for small solutions near onset to an equivalent Hamiltonian system with finitely many degrees of freedom. Normal-form theory yields the existence of homoclinic solutions to the reduced system, which correspond to spatially localised solutions of the ferrohydrostatic equations.