Fluid-stochastic models to describe the dynamics of the Faraday pilot waves in the long memory regime

For a suitably vibrating bath of fluid, a small droplet of the same fluid will "walk" across the surface due to the propulsive interactions with the waves generated by the previous droplet-bath impacts. As the bath vertically vibrates at a sufficiently large (yet subcritical) amplitude, the droplet will bounce periodically. The bouncing droplet does not make contact with the bath, it is instead propelled back into the air due to the cushioning effect of the lubrication layer of air trapped between the bath and droplet visible only on a microscopic scale. At each impact the droplet triggers a wavefield consisting of propagating and Faraday waves. As the forcing amplitude increases, the triggered waves increase in amplitude and the droplet destabilises and receives a "kick" in the horizontal direction. This results in the drop walking along the surface. Increasing the forcing vibration will increase the Faraday wave's decay time yielding a longer path "memory" from previous drop impacts.

The main object of study, a Faraday pilot wave is the pair consisting of the droplet (particle) and the associated Faraday wave. Their dynamics are complex (chaotic) and non-local in time (i.e. there is memory in the system). The goal of the present research is to develop hybrid fluid-stochastic models to describe the dynamics of the Faraday pilot waves in the long memory regime. This will enable the understanding of certain hydrodynamic quantum analogues that have been experimentally discovered, such as the wavelike statistics when particles are confined to a corral, double-quantization when particles are confined by a harmonic potential, tunnelling across barriers, and single- and double-slit diffraction. In particular, an important open problem is the equation governing the probability distribution for particles.

The research will involve:
1.Modelling the Faraday pilot wavefield;
2.One-dimensional and three-dimensional analysis of the system;
3.Numerical wave simulations;
4.Stochastic modelling of the system;
5.Stochastic Differential Equation (SDE) analysis including McKean-Vlasov processes;
6.Stochastic simulations;
The supervisory team consists of Prof. Paul Milewski who is an expert on physical modelling of fluid and continuum processes and on wave dynamics, and Prof. Tim Rogers who is an expert on stochastic differential equations and stochastic modelling. Both have computational expertise in their fields. Together they span the expertise needed for the project.

The impact of the SAMBa CDT will occur principally through the following two pathways:

1. Direct engagement with industrial partners, leading to PhD projects that are collaborative with industry, and that are focussed on topics with direct industrial impact.

2. The production of PhD graduates with
(a) the mathematical, statistical and computational technical skill sets that have been identified as in crucial demand both by EPSRC and by our industrial partners, coupled to
(b) extensive experience of industrial collaboration.

The underlying opportunity that SAMBa provides is to train graduates to have the ability to combine complex models with 'big data'. Such people will be uniquely equipped to deliver impact: whether they continue with academic careers or move directly to posts in industry, through quantitative modelling, they will provide the information that gives UK businesses competitive advantages. Our industrial partners make it clear to us that competitiveness in the energy, manufacturing, service, retail and financial sectors is increasingly dependent on who can best and most quickly analyse the huge datasets made available by the present information revolution.

During their training as part of SAMBa, these students will have already gained experience of industrial collaboration, through their PhD projects and/or the Integrated Think Tanks (ITTs) that we propose, that will give all SAMBa students opportunities to develop these transferable skills. PhD projects that involve industrial collaboration, whether arising from ITTs or not, will themselves deliver economic and social benefits to UK through the private companies and public sector organisations with which SAMBa will collaborate.

We emphasise that Bath is at the forefront of knowledge transfer (KT) activities of the kind needed to translate our research into impact. Our KT agenda has recently been supported by KT Accounts and Impact Acceleration Accounts from EPSRC (£4.9M in total) and a current HEFCE HEIF allocation of £2.4M. Bath is at the forefront of UK activity in KTPs, having completed 150 and currently holding 16 KTP contracts worth around £2.5M.

The SAMBa ITTs are an exciting new mechanism through which we will actively look for opportunities to turn industrial links into research partnerships, supported in the design of these projects by the substantial experience available across the University.

More widely, we envisage impact stemming from a range of other activities within SAMBa:

- We will look to feed the results of projects involving ecological or epidemiological data directly into environmental and public health policy. We have done this successfully many times and have three REF Case Studies describing work of this nature.

- Students will be encouraged to make statistical tools available as open source software. This will promote dissemination of their research results, particularly beyond academia. There is plenty of recent evidence that such packages are taken up and used.

- Students will discuss how to use new media to promote the public understanding of science, for example contributing to projects such as Wikipedia.

- Students will be encouraged to engage in at least one outreach activity. Bath is well known for its varied, and EPSRC-supported, public engagement activities that include Royal Institution Masterclasses, coaching the UK Mathematics Olympiad team, and reaching 50 000 people in ten days with an exhibit at the Royal Society's 350th Anniversary Summer Exhibition in 2010.