My research project
Localised Radial Spots on the Surface of a Ferrofluid
I am investigating the emergence of localised axisymmetric (radial) spots on the surface of a ferrofluid in the Rosensweig instability experiment. This work is under the supervision of Prof David Lloyd and Dr Matthew Turner and is an EPSRC funded PhD studentship.
Ferrofluids are a mixture of iron particles within a carrier fluid, causing the free surface of this fluid to be manipulated when an magnetic field is applied. Patterns emerge, such as spots or a hexagonal array of ridges and proving the existence of these patterns mathematically is a complex process. Localised radial spots, single axisymmetric peaks that emerge from the flat ferrofluid, have been shown to exist experimentally, by locally increasing the magnetic field in the fluid. These spots persist after the field is reduced, and so it remains a mystery if the existence of these spots is related to an underlying mechanism where they can spontaneously emerge from the flat state. The main objectives of the project are to rigorously prove the existence of localised radial spots in the ferrofluid problem using mathematical analysis to determine whether the spot emerges from the flat surface state or another state. The novelty of the present project lies in the using of currently developed techniques, such as radial centre-manifold reduction, asymptotic techniques and linear stability theory to answer the above questions. However, this is non-trivial as some of the above mentioned techniques need to be developed further to incorporate the geometry of the problem.
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)
- LMS Early Research travel grant, value £500, London Mathematical Society- for travel to SIAM NWCS20*
- FEPS travel award, value £850, Faculty of Engineering and Physical Sciences, University of Surrey- for travel to SIAM NWCS20*
*Conference cancelled due to the COVID-19 pandemic
- FEPS travel award, value £770, Faculty of Engineering and Physical Sciences, University of Surrey- for travel to Equadiff 2019
Talks and Presentations
- Bridging the Gap: the Hunt for Localised Hexagonal Patches, 15th October 2020 - Postgraduate Research Seminars (Internal Research Seminar), University of Surrey
- Mountains out of Magnets - Existence of Localised Radial Peaks on the Surface of a Ferrofluid, 3rd June 2020 - Waves in One World (Online Seminar Series)
- Mountains out of Magnets - Existence of Localised Radial Patterns on the Surface of a Ferrofluid, 28th May 2020 - Chair for Analysis and Modelling Seminar (Online Research Seminar), University of Stuttgart
- Magnetic Mountains - Investigating the Existence of Localised Axisymmetric Patterns on the Surface of a Ferrofluid, 5th May 2020 - Applied Analysis Seminar (Online Research Seminar), University of Bremen
- Magnetic Mountains - Investigating the Existence of Localised Axisymmetric Patterns on the Surface of a Ferrofluid, 30th April 2020 - Joint NWGFD & DSPDEs Seminar (Research Seminar), University of Surrey
- Localised Radial Peaks on the Surface of a Ferrofluid, 8th January 2020 - Surface and Internal Waves SIG (Contributed Talk), University of Surrey
- Localised Spots on a Ferrofluid, 9th July 2019 - Equadiff 2019 (Poster Presentation), Leiden
- Localised Radial Solutions on the Surface of a Ferrofluid, 4th July 2019 - University of Saarlandes Applied Analysis Seminars (Research Seminar), Saarbrücken
- Localised Radial Spots on the Free Surface of a Ferrofluid, 24th April 2019 - British Applied Mathematics Colloquium 2019 (Contributed Talk), Bath
- Localised Spots on the Surface of a Ferrofluid, 21st February 2019 - Postgraduate Research Seminars (Internal Research Seminar), University of Surrey
- > Just Numbers?, 13th March 2020 - Duke of Kent School Pi Day Mathematics Event (Mathematics Workshop), Duke of Kent School, Ewhurst
- Peaks and Troughs: Researching Patterns on the Surface of a Magnetic Fluid, 6th November 2019 - Mathematics PhD Open Afternoon 2019 (Research Talk), University of Surrey
- Is Maths Relevant in the Real World?, 22nd July 2019 - Maths WP Summer School 2019 (Research Project Introduction), University of Surrey
- Population Modelling or, How I Learned to Stop Worrying and Plot the Graph, 2nd July 2019 - Royal Alexandra and Albert School Campus Visit (Maths Taster Talk), University of Surrey
- Booms and Chaos: The Perils of Population Modelling, 20th March 2019 - Mathematics Taster Day (Applied Mathematics Workshop), University of Surrey
- Pattern Formation in Ferrofluids: What rocket fuel can teach us about geometric structures in nature, 31st October 2018 - Mathematics PhD Open Day 2018 (Research Talk), University of Surrey
- Why is Nature so Unoriginal? An Introduction to the Study of Pattern Formation, 30th October 2018 - `Taste of Research' Undergraduate Seminars (Research Seminar), University of Surrey
- Maths in the Real World: An Introduction to Study and Research at Surrey, 9th July 2018 - Maths WP Summer School 2018 (Research Project Introduction), University of Surrey
Postgraduate Research Seminars (PRS)
These seminars by PhD students are given to an audience of PhD students and some undergraduates, providing a more casual environment to present ideas and practise presentation skills. The talks are designed to introduce a student's area of research/interest and create a more inter-connected mathematics community. For more information, please contact Kieran Boniface or Imran Usmani.
- 11 Oct 2018 - Ying Huang: `Lagrangian fluid dynamics'
- 25 Oct 2018 - Dónal Harkin: `Bayesian approaches to estimating the flow field of water, induced by burrowing lugworms, in marine sediments'
- 8 Nov 2018 - Sean Cleator: `Reanalysis of ice-age paleoclimates using data asssimilation'
- 22 Nov 2018 - Jane Lyle: `Introduction to Machine learning'
- 13 Dec 2018 - Jacob Brooks: `Fronts in Inhomogeneous Wave Equations'
- 17 Jan 2019 - Roberto Sisca: `Where do spinors come from?'
- 31 Jan 2019 - Michael Foskett: `Geometry of quantum hydrodynamics and applications to quantum chemistry'
- 21 Feb 2019 - Dan Hill: `Localised spots on the surface of a ferrofluid'
- 7 March 2019 - Rebecca Atkinson: `Localisation and Optimal Mitigation of Sampling Error in Ensemble Data Assimilation'
- 21 March 2019 - Tom O'Neill: `Large Scale Atmospheric Flows and the Semigeostrophic Equations'
- 4 April 2019 - Josephine Solowiej-Wedderburn: `Sticking around: models for cellular force generation and the cell-substrate interface'
- 2 May 2019 - Nick Burgess: `Modulation using Group Theory'
- 17 Oct 2019 - Carl Haines (University of Reading): `Forecasting Geomagnetic activity with an Analogue Ensemble and Support Vector Machine'
- 31 Oct 2019 - Tommaso Macrelli: `Scattering amplitude recursion relations and L-infinity algebras'
- 14 Nov 2019 - Marcel Dengler: `Status update on the Navier-Stokes Millenium Problem'
- 23 Jan 2020 - Jane Lyle: `Attractor Reconstruction of Periodic Signals'
- 6 Feb 2020 - Jonah Varney: `Ergodic Theory and some applications'
- 20 Feb 2020 - Nick Winstone (Royal Holloway University): `Variants of Thompson's Groups F, T, and V'
- 5 Mar 2020 - Imran Usmani: `Mathematical Modelling of the Circadian Rhythm'
- 15 Oct 2020 - Dan Hill: `Bridging the Gap: The Hunt for Localised Hexagonal Patches'
`Taste of Research' Undergraduate Seminars (TORUS)
These seminars by PhD students are designed to introduce year 2, 3 and 4 mathematics students to a taste of mathematics doctoral research. For more information regarding the `Taste of Research' Undergraduate Seminars click here, or contact Steve Falconer.
I completed my MMath project under the supervision of Dr Bin Cheng (University of Surrey), rigorously investigating solutions to the Navier-Stokes equations in a thin spherical shell. This involved a careful application of functional analysis in a small, non-convex domain, where we identified a collection of analytical problems to overcome in future work.
Following this, I began to work with Prof David Lloyd and Dr Matthew Turner in the areas of dynamical systems and nonlinear waves, where I am currently an active member of the Nonlinear Waves and Geometric Fluid Dynamics and the Dynamical Systems and PDEs research groups.
My current research focuses around the study of localised patterns on the surface of a ferrofluid, a magnetic fluid consisting of iron nanoparticles. In 2005, for a vertically applied magnetic field, localised radial spikes were experimentally observed emerging from the flat state.
In order to find localised solutions to this non-autonomous PDE system, which can be formulated as a variational problem with Hamiltonian structure in the far-field, we decomposed solutions onto a basis which is independent of the radius. This reduces the problem to an infinite set of nonlinear, non-autonomous ODEs. Using radial centre manifold theory, we constructed local manifolds of small-amplitude solutions in the core and far-field regions. Then, we introduce geometric blow-up coordinates to track localised radial solutions through these manifolds and determine their radial profile.
We found three classes of stationary radial solutions: elevated and depressed spots which have a larger magnitude at the core, and rings which have algebraic decay towards the core. Notably, these correspond exactly to the classes of radial solutions found for the Swift-Hohenberg equation (SHE), a prototypical pattern-forming ODE model. In fact, we discovered that the centre-modes of the radial ferrofluid problem reduce completely to the radial SHE and so, for more complicated problems, it makes sense to study this simpler model first.
Recently we have begun to investigate the existence of localised hexagonal patches in the two-dimensional polar SHE. By decomposing the problem into a polar Fourier approximation, for a finite order N, we are able to treat the SHE as N coupled radial ODEs. Using numerical continuation codes in MATLAB, we were able to find solutions for a localised N-layered hexagonal patch, as well as identify cascading bifurcations as the bifurcation parameter approaches zero, where new layers of the patch emerge at each bifurcation. Using the previously established theory for radial problems, we are currently attempting to analytically prove the existence of an N-layered hexagonal patch and show that it becomes a full hexagonal lattice in the limit as N tends to infinity.