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

TL;DR

This paper proves the existence of static, doubly periodic surface patterns on a ferrofluid near the onset of the Rosensweig instability, using advanced mathematical techniques for nonlinear boundary-value problems.

Contribution

It introduces a novel formulation of ferrohydrostatic equations via Dirichlet-Neumann operators and applies bifurcation theory to analyze pattern formation.

Findings

Existence of static periodic patterns near instability threshold

Analytic characterization of bifurcation types

Development of a new mathematical framework for ferrofluid surface analysis

Abstract

We establish the existence of static doubly periodic patterns (in particular rolls, rectangles and hexagons) on the free surface of a ferrofluid near onset of the Rosensweig instability, assuming a general (nonlinear) magnetisation 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.