Localised radial spots on the free surface of a ferrofluid

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. Further extensions of the project once the above questions have been answered could include looking at how two or more of these spots interact with each other once they form.

This project fits into the EPSRCs Mathematical Sciences Theme under the research areas of Mathematical Analysis and Continuum Mechanics