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Dan Hill

Postgraduate Research Student

Academic and research departments

Department of Mathematics.

My research project

University roles and responsibilities

  • Maths Department PG/R Rep (2018- 20)
  • Founder & Coordinator of `Taste of Research' Undergraduate Seminars (2018-20) [See Talks and Presentations Section]
  • Founder & Coordinator of `Postgraduate Research Seminars' (2018-20) [See Talks and Presentations Section]
  • Postgraduate Student Mentor (2018-19)

    My qualifications

    MMath (Mathematics) - 86 per cent (1st Class Award)
    University of Surrey


    Research interests

    My publications


    Dan J. Hill, David J.B. Lloyd, Matthew R. Turner (2020). Localised Radial Patterns on the Free Surface of a Ferrofluid (Submitted)
    View abstract View full publication
    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-radius cylinder 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. Four distinct classes of stationary radial solutions are found: elevated and depressed spots, which have a larger magnitude at the core, and elevated and depressed rings, which have algebraic decay towards 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 establishing the existence of axisymmetric localised patterns in the future.