Modelling, computation and analysis of droplets guided by Faraday waves: a complex system with macroscopic quantum analogies.
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
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.
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.
Planned Impact
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.
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.
Organisations
- University of Bath (Lead Research Organisation)
- University College London (Collaboration)
- Brown University (Collaboration)
- University of Warwick (Collaboration)
- Massachusetts Institute of Technology (Collaboration, Project Partner)
- Institute for Pure and Applied Mathematics (Collaboration)
- Instituto Nacional de Matemática Pura e Aplicada (Project Partner)
People |
ORCID iD |
Paul Milewski (Principal Investigator) |
Publications
Bush JWM
(2018)
Introduction to focus issue on hydrodynamic quantum analogs.
in Chaos (Woodbury, N.Y.)
Damiano A
(2016)
Surface topography measurements of the bouncing droplet experiment
in Experiments in Fluids
Durey M
(2018)
Dynamics, emergent statistics, and the mean-pilot-wave potential of walking droplets.
in Chaos (Woodbury, N.Y.)
Durey M
(2020)
Faraday pilot-wave dynamics in a circular corral
in Journal of Fluid Mechanics
Durey M
(2017)
Faraday wave-droplet dynamics: discrete-time analysis
in Journal of Fluid Mechanics
Galeano-Rios C
(2018)
Ratcheting droplet pairs
in Chaos: An Interdisciplinary Journal of Nonlinear Science
Galeano-Rios C
(2017)
Non-wetting impact of a sphere onto a bath and its application to bouncing droplets
in Journal of Fluid Mechanics
Galeano-Rios C
(2021)
Capillary-scale solid rebounds: experiments, modelling and simulations
in Journal of Fluid Mechanics
Galeano-Rios C
(2019)
Quasi-normal free-surface impacts, capillary rebounds and application to Faraday walkers
in Journal of Fluid Mechanics
Gao T
(2019)
Hydroelastic solitary waves with constant vorticity
in Wave Motion
Description | This research has yielded fundamental results in the mathematical modelling and theoretical understanding of the Faraday pilot wave problem. Faraday waves are waves generated by the vertical vibration of a vessel. It was discovered by the physicist Yves Couder that droplets may bounce indefinitely off a vibrating fluid surface and that at each bounce they generate a Faraday wavefield. This Faraday wavefield then provides a "landscape" which guides the drop horizontally. This is a remarkable macroscopic analogy to particle-wave duality and several classic quantum mechanical experiments have analogues in this fluid system. We have obtained several fundamental results: (1) Formulated a "discrete impact map" that models the problem from basic fluid mechanical principles; (2) Rationalised the experimentally observed double quantization of a particle in a harmonic potential using combined statistical and fluid methods; (3) Discovered the importance of a "mean-pilot-wave potential" which is generated by confined particles and governs their statistical behaviour; (4) Simulated tunnelling over a submerged barrier and more generally developed a framework for dealing with Faraday waves and submerged topography; (5) Developed a fundamental model of solid sphere and free surface impact and rebound at capillary scales; (6) Collaborated with experimentalists and CFD experts to validate various rebounding particle problems. |
Exploitation Route | The outcomes have huge potential, ranging from fundamental questions in quantum mechanics to applications of the methods developed for practical simulations such as droplet generation in medical devices (nebulizers) by Faraday waves. |
Sectors | Environment,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Other |
Description | Royal Society International Exchanges Cost Share |
Amount | £9,250 (GBP) |
Funding ID | IEC\R2\170195 |
Organisation | The Royal Society |
Sector | Charity/Non Profit |
Country | United Kingdom |
Start | 03/2018 |
End | 02/2020 |
Description | Brown - Daniel Harris |
Organisation | Brown University |
Country | United States |
Sector | Academic/University |
PI Contribution | Collaboration on experiments of superhydrophobic spheres. |
Collaborator Contribution | Experimental investigations on impacting and rebound of superhydrophobic spheres. |
Impact | Paper in preparation |
Start Year | 2017 |
Description | IMPA- Prof. Andre Nachbin |
Organisation | Institute for Pure and Applied Mathematics |
Country | United States |
Sector | Academic/University |
PI Contribution | We are the foremost computational group studying Faraday wave-droplet interaction and we collaborate by sharing theoretical and numerical results and methods with the MIT group. We also hate theoretical, numerical and methodological results for other types of water wave problems: waves with vorticity and interaction of waves and currents. |
Collaborator Contribution | The partners provide technical contributions in the area of applications of conformal maps and by providing studentship, post-doctoral and conference support to projects. |
Impact | See joint author publications |
Start Year | 2012 |
Description | MIT- Prof. John Bush |
Organisation | Massachusetts Institute of Technology |
Department | Department of Mathematics |
Country | United States |
Sector | Academic/University |
PI Contribution | We are the foremost computational group studying Faraday wave-droplet interaction and we collaborate by sharing theoretical and numerical results and methods with the MIT group. |
Collaborator Contribution | The MIT group of Prof. John Bush is the foremost group in the world in research on Faraday-droplet experiments. They collaborate by sharing experimental and modelling results and hosting out team. |
Impact | See publications - several have co-authors from both groups. |
Start Year | 2014 |
Description | UCL - Prof. J-M Vanden-Broeck |
Organisation | University College London |
Department | Mathematics |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | This partnership involves the complementary analysis/numerical skills between Milewski's group at Bath and Vanden-Broeck's group at UCL. Milewski brings experience in time dependent problems, spectral methods and internal waves. |
Collaborator Contribution | This partnership involves the complementary analysis/numerical skills between Milewski's group at Bath and Vanden-Broeck's group at UCL. Vandal-Broeck brings experience in nonlinear free-boundary steady problems and numerical methods. |
Impact | See joint publications. |
Start Year | 2011 |
Description | Warwick - Radu Cimpeanu |
Organisation | University of Warwick |
Country | United Kingdom |
Sector | Academic/University |
PI Contribution | Full fluid simulations of solid sphere - fluid impact at capillary scales. |
Collaborator Contribution | Designed and ran simulations. |
Impact | Paper in preparation |
Start Year | 2018 |
Description | Research in the Press - O Globo January 2017 (in Portuguese) |
Form Of Engagement Activity | A press release, press conference or response to a media enquiry/interview |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Media (as a channel to the public) |
Results and Impact | This was a major interview (1/2 newspaper page) in printed and web press by on of the top 3 newspapers in Brazil on various aspects of my research but mainly Quantum Hydrodynamic Analogues. |
Year(s) Of Engagement Activity | 2017 |
URL | http://oglobo.globo.com/sociedade/conte-algo-que-nao-sei/paul-milewski-matematico-os-tsunamis-vao-co... |