Small-amplitude static periodic patterns at a fluid-ferrofluid interface
arXiv:1801.08478 · doi:10.1098/rspa.2018.0038
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.
23 pages, 9 figures