Colloquium: Small-amplitude cellular patterns at a fluid-ferrofluid interface, Mark Groves (Universität des Saarlandes)
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The Rosensweig instability is observed experimentally when a vertical magnetic field is applied to a static ferrofluid layer. Regular cellular patterns (rolls, squares or hexagons) appear on the ferrofluid surface as the field strength is increased through a critical value.
Despite a wealth of experimental and numerical eveidence, until recently very little was known theoretically on the existence of solutions to the governing equations for this ferrohydrostatic problem. Progress has however been made by exploiting similarities with the classical problem for travelling water waves.
In this talk we discuss a formulation of the ferrohydrostatic equations in terms of Dirichlet-Neumann operators for nonlinear elliptic boundary-value problems. The operators are shown to be analytic in a certain sense, and the Rosensweig instability is studied using an analytic version of the celebrated Crandall-Rabinowitz local bifurcation theory.