Abstract
This paper investigates the existence of localised axisymmetric (radial) patterns on the surface of a ferrofluid in the presence of a uniform vertical magnetic field. We formally investigate all possible small-amplitude solutions which remain bounded close to the pattern’s centre (the core region) and decay exponentially away from the pattern’s centre (the far-field region). The results are presented for a finite-depth, infinite expanse of ferrofluid equipped with a linear magnetisation law. These patterns bifurcate at the Rosensweig instability, where the applied magnetic field strength reaches a critical threshold. Techniques for finding localised solutions to a non-autonomous PDE system are established; solutions are decomposed onto a basis which is independent of the radius, reducing the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, local manifolds of small-amplitude solutions are constructed in the core and far-field regions, respectively. Finally, using geometric blow-up coordinates, we match the core and far-field manifolds; any solution that lies on this intersection is a localised radial pattern. Three distinct classes of stationary radial solutions are found: spot A and spot B solutions, which are equipped with two different amplitude scaling laws and achieve their maximum amplitudes at the core, and ring solutions, which achieve their maximum amplitudes away from the core. These solutions correspond exactly to the classes of localised radial solutions found for the Swift–Hohenberg equation. Different values of the linear magnetisation and depth of the ferrofluid are investigated and parameter regions in which the various localised radial solutions emerge are identified. The approach taken in this paper outlines a route to rigorously establish the existence of axisymmetric localised patterns in the future.
Highlights
Ferrofluids are liquids with a colloidal suspension of magnetic nanoparticles; as a result, they can be manipulated by the application of an external magnetic field; see Torres-Diaz and Rinaldi (2014) and Rosensweig (2013) for a review of ferrofluids and their associated experiments)
A quiescent layer of ferrofluid in a flat, horizontal container is subjected to a uniform vertical magnetic field of magnitude h, see Fig. 1a
Theoretical Approaches to Ferrofluids The first attempt at an analytic study of domain-covering cellular patterns of the Rosensweig instability was by Gailıtis (1977), who substituted a cellular free-surface ansatz into a hypothetical free energy for the system
Summary
Ferrofluids are liquids with a colloidal suspension of magnetic nanoparticles; as a result, they can be manipulated by the application of an external magnetic field; see Torres-Diaz and Rinaldi (2014) and Rosensweig (2013) for a review of ferrofluids and their associated experiments). The spikes emerged away from the centre of the container were able to be moved around the container, and did not appear to feel the effects of the container boundary This suggests the spots are well-localised, possibly decaying exponentially fast to the flat state. Theoretical Approaches to Ferrofluids The first attempt at an analytic study of domain-covering cellular patterns of the Rosensweig instability was by Gailıtis (1977), who substituted a cellular free-surface ansatz into a hypothetical free energy for the system (an infinite-depth ferrofluid with a linear magnetisation law) This resulted in finding regions of existence for various lattice patterns (squares, stripes and hexagons), and was later extended to finite-depth ferrofluids by Friedrichs and Engel (2001). Prototypical pattern forming systems have been analysed to prove the existence of localised radial solutions to the Swift–Hohenberg
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