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
- Athena Swan committee member (2020-21)
- 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)
Prizes & Awards
- Finalist: IMA Lighthill Thwaites Prize
- Awarded: First Year Departmental Award for Excellent Performance
- 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
- Localised Radial Patterns on the Free Surface of a Ferrofluid, 22nd-27th August 2021 - International Congress of Theoretical and Applied Mechanics 2020+1 (Oral Presentation)
- The Existence of Localised Radial Patterns on the Surface of a Ferrofluid, 24th May 2021 - SIAM Conference on Applications of Dynamical Systems 2021 (Contributed Talk)
- Making Mountains out of Magnets: Localised Patterns on the Surface of a Ferrofluid, 11th May 2021 - Leeds Applied Nonlinear Dynamics Seminars (Online Research Seminar), University of Leeds
- On the existence of localised radial patterns on the surface of a ferrofluid, 6th April 2021 - British Applied Mathematics Colloquium 2021 (IMA Lighthill-Thwaites Prize Minisymoposium), University of Glasgow
- 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
- Something from nothing: Investigating the emergence of localised patterns, 2nd December 2020 - `Taste of Research' Undergraduate Seminars (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 existence of localised patterns; in particular, I investigated localised radial 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.
The ferrofluid experiment can be formulated as a free-surface problem, which we expressed as a quasilinear PDE system that is non-autonomous in the radial coordinate r. We constructed an r-independent basis formed from eigenmodes of the linear operator in the far-field; projecting onto each respective mode reduces the PDE system to an infinite set of nonlinear radial ODEs. Using the theory of stable foliations and local invariant manifolds, we constructed local manifolds in the core and far-field regions, containing all solutions that remain bounded in the core and decay exponentially in the far-field, respectively. Using geometric blow-up coordinates seen for the Swift-Hohenberg equation (SHE) and looking for intersections of the core and far-field manifolds, we identified three classes of localised radial patterns and determined their radial profiles. Notably, these solutions correspond exactly to the localised radial patterns found in the SHE and one of these solutions, known as spot A, is believed to be the spot observed in the ferrofluid experiment.
Another interesting phenomena is the emergence of localised cellular patterns; such as rhomboids, square, and hexagons, for example. Localised hexagons have been observed bifurcating from the radial spot in the ferrofluid experiment, and a large variety of localised cellular patterns have been found numerically and experimentally in other areas of research; for example, in nonlinear optics.
We have recently been investigating the existence of localised cellular patterns in the SHE, since it represents one of the simplest pattern-forming systems. We approximated the full planar SHE by a truncated Fourier series in polar coordinates; by projecting onto each Fourier mode, we obtain a finite-dimension coupled system of nonlinear radial ODEs. Then, extending the techniques seen for radial solutions, we again construct core and far-field manifolds and look for intersections. In this case, localised solutions are subject to an algebraic matching condition that depends on the truncation order N and lattice m imposed by our Fourier decomposition. By solving the small truncation matching problems (N=1,2,3), we are able to use numerical continuation codes in MATLAB to find larger localised cellular patterns in parameter space.