Modelling, computation and analysis of droplets guided by Faraday waves: a complex system with macroscopic quantum analogies.

When a fluid filled container is shaken vertically, one may observe waves on the surface of that container if the shaking is sufficiently strong. These waves arise out of a subharmonic instability: they have half the frequency of the shaking, and are called Faraday waves.

In separate experiments, high-speed films of droplet impacts on static fluid baths show that the droplet does not always coalesce with the bath on impact, but may bounce a few times before coalescing.

Combining these two experimental facts, about 10 years ago it was discovered that millimetric liquid droplets can bounce indefinitely when dropped on the surface of a bath of the same liquid in a shaken container. The phenomenon occurs below the shaking threshold for the Faraday instability. More surprisingly, the droplets can also spontaneously "walk" along the surface of the vibrating bath. These walkers then exhibit many features previously thought to be exclusive to quantum mechanics such as wave-particle duality, quantised energy states, single particle diffraction and tunnelling behaviour.

The aim of this proposal is to explore the fluid mechanical aspects of this system. There are many unanswered challenging questions due to the complexity of the problem. Fluid mechanics questions include the understanding of non coalescing drop impact, and the behaviour of reflecting walkers and their pilot wave field at walls. We will also seek to understand how a purely classical mechanics system exhibits quantum mechanical-like behaviour, and probe the limits of this analogy.

The proposed research involves a combination of analytical and numerical approaches as well as comparisons with experiments. We will partner with an MIT state-of-the-art fluid dynamics laboratory which will provide both experimental data and design validation experiments. The problems we plan to study are of general interest in fluid mechanics and in the theory of free boundary problems and dynamical systems. It is expected that the results will have broad applications, in particular to the understanding of the impact of drops and particles with fluids.

Faraday instabilities are also the most reliable way of generating consistently sized droplets continuously. Because of this there are several possible microfluidics applications for this research, such as developing better devices for delivering inhaled drugs.

The work proposed is aligned with the EPSRC priority of Fluid Mechanics, and, within that area, the modelling and simulation of fluid surface interfaces across scales and in particular at small scales.

Free boundary fluid problems have far reaching applications both in the study of our environment and on many aspects of human activity. Their mathematical study has a broad range of impacts beyond the field of mathematics. For example, understanding capillary gravity waves - which are the waves studied in this research - are an important ingredient in satellite remote sensing of the ocean's surface, and used to infer surface wind conditions from surface roughness. Rain droplet impact and sea spray generation is a major contributor to air-ocean chemical exchanges.

Parametrically forced Faraday waves, also have many applications from the microfluidic behaviour of acoustically driven bubbles in medical applications (Leighton 2004), to dynamics of liquid surfaces in space conditions (Ibrahim 2005). In particular the Faraday instability of a liquid surface is the most efficient way of producing a continuous stream of droplets of a required size. This has applications in economically important applications such as combustion engines (Boukra et al 2009) and medical devices. An example of a recent application is in the development of new medical nebuiliser devices for pulmonary drug delivery which can produce a continuous stream of consistently sized micrometer drops reliably (Tsai et al 2014).

Our research will have a strong human resource impact component. We will employ and train a highly skilled post-doctoral research associate (PDRA) who will gain new training and practice in cutting edge fluid mechanics of free-boundary problems and in the numerical simulation of such problems. The PDRAs research and training will be enhanced three monthlong placements in a state-of-the-art lab at the Massachusetts Institute of Technology (MIT). The research project will also benefit doctoral students and train them similarly, in highly desirable skills. At Bath one student associated to the SAMBa CDT has expressed the interest of joining the project, and at UCL, Prof. Vanden-Broeck has an extensive track record of PhD mentoring. PhD students at UCL and Bath will have the benefit of interacting with the project's PDRA and interaction with the visiting scientists and project partners. PhD student activity is complementary to the proposal and is funded by other means. The PDRAs and students will be part of a much larger community of applied mathematicians and benefit, through the attendance to seminars and other activities (such as the SAMBa Integrated Think Tanks), of a broad exposure to applied, computational and industrial mathematics.


REFERENCES:

R. A. Ibrahim Liquid Sloshing Dynamics: Theory and Applications (2005), Cambridge University Press.

T. G. Leighton (2004), From seas to surgeries, from babbling brooks to baby scans: the acoustics of gas bubbles in liquids, Int. J. Mod. Phys. B, 18, 3267.

Madjid Boukra, Alain Cartellier, Éric Ducasse, Pierre Gajan, Marie Lalo, Thomas Noel, Alain Strzelecki (2009) Use of Faraday instabilities to enhance fuel pulverisation in air-blast atomisers, Comptes Rendus Mécanique, 337, 492-503.

C.S. Tsai, R.W. Mao, S.K. Lin, S.C. Tsai, (2014) Faraday instability-based micro droplet ejection for inhalation drug delivery, TECHNOLOGY, vol. 2, pp. 75-81.