Exponential asymptotics for multi-dimensional systems in fluid mechanics

Lead Research Organisation: University of Bath
Department Name: Mathematical Sciences

Abstract

The vast majority of problems that lie at the forefront of science are governed by mathematical equations that cannot be solved exactly. In the modern era, large-scale numerical computation and data analysis are powerful tools, but many questions still elude brute-force computation. For complex multi-scale and multi-parameter systems, it is often necessary to apply key reductions dependent on the smallness or largeness of certain parameters. The application of these reductions is called asymptotic analysis; these methods have the power to dramatically simplify complex systems to their salient features, extract key mechanisms, and provide details in regions where numerics and experiments fail. As noted by Crighton [1] "[the] design of computational or experimental schemes without the guidance of asymptotic information is wasteful at best, and dangerous at worst, because of the possible failure to identify crucial (stiff) features..."

Some of the most challenging problems relate to the prediction of exponentially small effects that are invisible to traditional asymptotic analysis and often mistakenly considered as negligible. In some cases, these effects may correspond to some observable feature, such as an oscillation or wave in the system; in other cases, they may be largely non-observable, but instead serve to determine whether certain solutions are permissible. Over the last few decades, there has been an appreciation for the ubiquity of problems where exponentially-small effects are paradoxically important -- these problems can be found in studies related to dendritic crystal growth, viscous fluid flow, water waves, quantum tunneling, geophysics, and more.

There are significant mathematical and computational challenges for the study of exponentially small terms. For example, the traditional mathematical techniques that exist, developed in the early 20th century, are usually insufficient. Exponential asymptotics is the name given to the set of specialised techniques that have been developed over the last two decades for these problems.

In the last few years, some of the most significant applications of exponential asymptotics have related to the development of theory for free-surface flows. This includes the study of (i) water waves produced by gravity-driven flows past slow-moving full-bodied ships; (ii) solitary waves in a fluid of finite depth including both gravity and capillary effects; and (iii) viscous flows where bubbles or fingers are produced at an interface. These problems all involve crucial exponentially small effects.

Despite the above successes, a significant bottleneck has emerged in numerous studies in the area: the majority of existing exponential asymptotic techniques are limited to ordinary differential equations where, for instance, only a one-dimensional fluid interface is considered. Many of the spectacular successes of exponential asymptotics that have emerged in the last two decades have analogues in higher-dimensional space or in time-dependent formulations, where the system is governed by partial differential equations. However, the standard techniques in exponential asymptotics are not easily adapted to study such situations.

The most recent preliminary work on seeking extensions of the theory has shown that the likely avenue for progress lies with combining analytical methods with computational and data-driven approaches---hence a hybrid numerical-asymptotic approach to exponential asymptotics. The development of these methodologies, and the subsequent applications to multi-dimensional problems in fluid mechanics forms the main thrust of this project.

[1] Crighton, D. G. (1994). Asymptotics--an indispensable complement to thought, computation and experiment in applied mathematical modelling. In Proc. 7th Eur. Conf. on Math. Industry (ECMI), Montecatini (pp. 3-19).

Publications

10 25 50
 
Description https://gtr.ukri.org/projects?ref=studentship-2427722


The intention of this was to develop sophisticated techniques in a specialised branch of mathematics known as exponential asymptotics. These specialised techniques allow the detection of small (exponentially-small) contributions that are paradoxically crucial in a number of areas in the physical sciences, notably fluid mechanics, water-wave interactions with bodies, the behaviour of thin wetting films, and viscosity-driven fluid mechanics. The work was motivated by the fact that, although such techniques of analysis have been developed previously for similar problems, they have been limited by virtue of being appropriate only for more limited mathematical models (notably models of one-dimensional and time-independent) phenomena. Therefore, the main thrust of the work undertaken related to extending these techniques so they could be applied to a much wider range of multi-dimensional problems in the physical sciences. In particular, the interest was also in scientific problems if this type where numerical approximations could be combined with analysis (since this is a key component of how such multidimensional extensions would work).

[1] Exponential asymptotics techniques with applications to geophysics. This part of the research programme was linked with the research associates Samuel Crew (SC) and Josh Shelton (JS). It was recognised early on that in order to handle the typical multi-dimensional systems that motivated the research programme, it was necessary to develop more general mathematical techniques that could tackle some of the known challenges. Together with SC, we developed a framework where techniques in adjacent areas known to theoretical physics (resurgence and Borel summation) could be re-framed and incorporated into more typical fluid mechanical or applied mathematics problems. Similarly, in projects with SC and JS, we investigated an important property known as "the higher-Stokes phenomenon", which is associated with the analysis of multi-dimensional problems. Together, these frameworks led to the development of mathematical methods for the study of instabilities in certain geophysical problems (notably the formation of Rossby and Kelvin waves in the atmosphere).

[2] Beyond-all-orders in wave-structure interactions. This part of the research programme led to additional PhD studentships at the University of Bath with Yyanis Johnson-Llambias (YJ-L), with funding provided by EPSRC and the Statistical Applied Mathematics Centre for Doctoral Training. With YJ-L, we were able to develop a computational and analytical methodology that indicated how water-waves were linked wave-generating bodies travelling in a three-dimensional fluid. Although this is a classic fluid mechanical problem, when the bodies move slowly in the water, the surface waves are challenging to compute; our analysis provides direct formulae for the wave-form and wave properties. Another significant result was the development of mathematics to study water-waves produced by flows past smoothed geometries. We demonstrated that prior methods of analysis were inadequate for dealing with wave-making geometries that were smoothed---our results again provide clear approximations for the wave properties.

[3] Beyond-all-orders in thin-film flows and viscous fingering. This part of the research programme led to additional PhD studentships at the University of Bath with Cecilie Andersen (CA), with funding provided by EPSRC and the Statistical Applied Mathematics Centre for Doctoral Training. Together with CA and additional researchers in Australia, we were able to develop a description of the fingering instability that occurs when an inviscid fluid is pumped into a thin channel filled with viscous fluid. In this scenario, the interface between inviscid and viscous fluids will typically go unstable, producing 'fingers'. Our techniques in exponential asymptotics were able to explain the formation of such fingers.
Exploitation Route All the problems studied in this research programme are connected to phenomena in the physical sciences, notably in areas of (i) water-waves and wave-structure interactions, and also in (ii) thin-film flows and viscous fluid dynamics. For example, in the first area (i), the outcomes of the funding are typically of interest to those in Engineering working in areas of computational fluid dynamics connected with oceanography and hydrodynamics. Although there have been many computational and modelling advances in this area, analytical solutions and approximations provide important tools for diagnosing and complementing numerical methods (e.g. benchmarking large-scale numerical experiments in regimes of low-speed waves). It is also important to recognise that this research programme centres upon techniques in asymptotic analysis, and this is widely used in all scientific industries, providing simple yet powerful estimates of solutions to ordinary differential or partial differential equations. The fundamental research has yielded important key results, but so too has been the training of the research associates and PhD students, of which two have held placement positions with impact (notably the UK Parliamentary Office of Science and Technology and also the Smith Institute).
Sectors Aerospace

Defence and Marine

Manufacturing

including Industrial Biotechology

 
Description The narrative impact of this award is still ongoing, and we expect to be able to report definitive impacts in late 2024-25. However, we can report the following. One of the key aims of this award was to bring the application of techniques in asymptotic analysis, and especially exponential asymptotics, to a wider audience. Asymptotic analysis is one of the key mathematical tools by which we can approximate solutions to certain problems (notably differential equations) which we would otherwise be required to compute numerically. As part of this award, we conducted a series of interviews with key researchers during a workshop in Okinawa, Japan, and during visits to Queensland and Sydney Australia. These interviews focused on 'humanising' elements of mathematical research, and probed at aspects like what a career in mathematics entails, what are the challenging problems to study, and the base intuition of much of the mathematician's research. These interviews have been recorded, and we will be looking to edit and process them in mid-to-late 2024. We will look forwards to sharing these videos with a broad audience through social media.
First Year Of Impact 2024
Sector Education
Impact Types Cultural

Societal

 
Description Exact asymptotics: from fluid dynamics to quantum gravity
Amount ¥10,000,000 (JPY)
Organisation Okinawa Institute of Science and Technology 
Sector Academic/University
Country Japan
Start 07/2023 
End 11/2023
 
Description Hybrid asymptotic-numerical schemes for exponentially small selection mechanisms
Amount £100,000 (GBP)
Funding ID 2427722 
Organisation University of Bath 
Sector Academic/University
Country United Kingdom
Start 09/2020 
End 03/2025
 
Description Multidimensional beyond-all-orders asymptotics for problems in fluid mechanics
Amount £100,000 (GBP)
Organisation University of Bath 
Sector Academic/University
Country United Kingdom
Start 09/2023 
End 10/2026
 
Description Pattern formation in continuous and discrete systems.
Amount £100,000 (GBP)
Funding ID 2284242 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 09/2019 
End 09/2023
 
Description Singular perturbation problems in wave-structure interactions
Amount £100,000 (GBP)
Funding ID 1940169 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 09/2017 
End 09/2021
 
Description Organiser and administrator for a series of public lectures on asymptotics and waves in Okinawa, Japan 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Public/other audiences
Results and Impact As part of an international visiting programme "Exact asymptotics: from fluid dynamics to quantum geometry" we worked together with the Okinawa Institute of Science and Technology to organise two public lectures that brought topics related to mathematics, physics wave theory, and dynamics chaos to both members of the general public and wider academic members in Japan and internationally. One talk in particular was about "Quantum Signatures of Chaos" and the search for chaotic dynamics in quantum mechanical systems and delivered by Prof. Akira Shudo. The second talk was about "Waves and Resonance: From Musical Instruments to Vacuum Cleaners, via Metamaterials and Invisibility Cloaks" and delivered by Professor Jon Chapman. The public talks generated much lively discussion afterwards.
Year(s) Of Engagement Activity 2023
URL https://groups.oist.jp/tsvp/exact-asymptotics-fluid-dynamics-quantum-geometry