Dan Hill is giving a virtual seminar via zoom to the Applied Analysis Group at the University of Bremen (link here), today 5th May. The title is “Magnetic Mountains — Investigating the existence of localised axisymmetric patterns on the surface of a ferrofluid“. Dan’s PhD project is supervised by Dave Lloyd and Matt Turner.
Abstract: Ferrofluids, magnetic fluids consisting of iron nanoparticles, provide a good experimental medium to investigate properties of nonlinear patterns. For a vertically applied magnetic field, there exists a critical field strength at which spikes emerge from the ferrofluid, arranging in domain-covering cellular patterns. In 2005, solitary spikes were experimentally observed; these spikes were not affected by the shape of the fluid’s container, and drifted around the domain, indicating they were localised solutions. In this talk, Dan will first highlight some of the techniques required to prove the existence of localised axisymmetric (radial) patterns for the prototypical Swift-Hohenberg ODE, including radial centre-manifold theory and geometric desingularisation. Following this, he will introduce the ferrohydrostatic problem, formulated as a PDE system, and present their formal results for showing the existence of localised radial patterns. This includes the introduction of an appropriate “spectral” decomposition, in order to reduce the problem to infinitely-many ODEs. Finally, we find that the ferrohydrostatic problem possesses the same classes of localised radial patterns as the Swift-Hohenberg equation, and explore the parameter regions in which these localised radial patterns emerge.