Nonlinear dynamics of microscale interfacial flows and model nonlinear partial differential equations

Lead Research Organisation: Imperial College London
Department Name: Department of Chemical Engineering

Abstract

This Overseas Travel Grant (OTG) proposal seeks funding to enable visits by the PI to Princeton U. and Tokyo U. of Science to undertake research in the following two subprojects:

1. Droplet motion with dynamic droplet variation (Princeton U.).

Droplet motion is ubiquitous in a wide spectrum of natural phenomena and technological applications. From the way various surfaces such as plant leaves and windows of our houses interact with rain droplets, to the rapidly growing field of micro- and nanofluidics. A problem of technological significance is that of polymer electrolyte membrane (PEM) fuel cells currently being investigated primarily experimentally at Princeton U. This system involves multiphase flows in complex geometries and in particular droplets that emerge from pores and grow into gas flow channels.

The full problem is quite involved and a simpler prototype, that of a droplet on a solid substrate with a small pore from which liquid can be pumped in or out (thus, emulating the growth process of droplets in the PEM cells), will be considered. Of particular interest are the effects of substrate disorder, either chemical or topographical and influence of noise. Indeed, in the PEM fuel cells the droplets are constantly in contact with disordered substrates and they are also subjected to fluctuations which are naturally present in the system. Despite its simplicity, this problem has a rather complex dynamics as we discuss in the Case of Support.

2. Nonlinear forecasting analysis of complex spatiotemporal behavior in spatially extended systems (SES) (Tokyo U. of Science).

SES are infinite-dimensional dynamical systems described through partial differential equations (PDEs) deterministic or stochastic in large or unbounded domains, and are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatiotemporal behavior. An example of such systems is the generalized Kuramoto-Sivashinsky (gKS) equation, a prototype that retains the fundamental elements of any nonlinear process that involves wave evolution in one dimension. The equation has been reported in a wide variety of physical and technological contexts, from plasma and geophysical phenomena to falling liquid films.

The deterministic gKS equation has received considerable attention over the years. One of the main findings is that sufficiently strong dispersion tends to regularise the spatiotemporal chaos of the KS equation in favor of spatially periodic cellular structures. The noisy gKS equation also appears in a wide variety of physical and technological contexts, e.g. evolution of solid films by sputtering. The proposed OTG seeks to explore the effects on noise on the gKS equation and to establish conditions under which it is possible to distinguish between the chaotic behavior of the gKS equation (for small dispersion) and the stochastic effects induced by noise.

A related problem is that of synchronization in noisy SES. Synchronization is central to many applications and natural phenomena, from electric circuits to biological systems, e.g. the cooperative behavior of living beings. Here we shall examine synchronization in noisy SES using a system of coupled noisy gKS equations as a prototype.

Planned Impact

It is useful to recall first the two Themes of the proposed research. Theme 1: Droplet motion with dynamic volume variation; Theme 2: Nonlinear forecasting analysis of complex spatiotemporal behavior of spatially extended systems (SES).

The economic and societal impact of the proposed research in Theme 1 will be realised through the development of modelling and predictive tools for small-scale complex fluid processes, crucial in a wide spectrum of technological applications, e.g. small-scale separations, lab-on-chip systems and microengineered devices in general. Such systems are key to several rapidly developing technologies in the UK's smart materials, specialty manufacturing sectors and biomedical industries. Future developments in these sectors will require sophisticated mathematical and computational techniques, as proposed here, which can contribute to many aspects, from improvements in product and design all the way to production.

The research will also lead to the development of state-of-the-art numerical methodologies for the accurate and reliable multiscale simulations of fluid flow in confined microgeometry. These codes will be of benefit to the control and optimisation of industrial processes and devices that exploit microscale flows.

A central aim of Theme 2 is to understand the effects of noise and synchronization in SES. This could be potentially crucial in several technological applications that can be approximated as noisy SES, e.g. evolution of thin solid films with sputtering and nanostructuring of solid surfaces by ion beam errosion. Such systems are key to several rapidly developing technologies in the UK, including smart materials and printing industries to name but a few. Understanding the effects of noise could offer the possibility to control their behavior, e.g. to stably confine particles in very small spaces, or to subject them to controlled forces. In this respect, noise could be used as an "engineering tool" to extract fundamental knowledge but also offer design solutions. Moreover, Theme 2 goes beyond technological applications, since noisy SES appear as model systems in a wide spectrum problems from biology to social dynamics.

Finally, the PI has a strong track record of knowledge transfer through close collaborations with industry, short courses, workshops and training of high-calibre researchers. The impact plan we have devised is based on a range of routes to maximise the likelihood of success and to reach as wide a community as possible: links with other projects and interactions with industrial partners through other projects, publication in leading journals, conference presentations, provision of a web-based dissemination and research colloquium.

Publications

10 25 50
publication icon
Duncan AB (2016) Noise-induced transitions in rugged energy landscapes. in Physical review. E

publication icon
Krumscheid S (2015) Data-driven coarse graining in action: Modeling and prediction of complex systems. in Physical review. E, Statistical, nonlinear, and soft matter physics

 
Description The present Overseas Travel Grant (OTG) application enabled the following two visits in 2015: (i) Profs Jay B. Benziger (JBB) and Yannis G. Kevrekidis (YGK) at the Chem. Biological Eng. Dept. of Princeton U. [YGK is also associated with the Program in Applied & Computational Maths (PACM) at Princeton],
and (ii) Prof. Hiroshi Gotoda (HG), a young talented academic at the Mech. Eng. Dept. of Tokyo U. of Science. Their expertise is as follows. JBB: chemical reaction engineering and multiphase systems; YGK: applied and computational maths, scientific computation of complex/multiscale systems modelling, nonlinear system identification and control; HG: dynamical systems analysis of complex phenomena in fluid mechanics and chemically reacting flows-combustion.

The proposed research divides into two subprojects/Themes each of which involves back-and-forth between theory and computations, fundamental and
applied aspects:
1. Droplet motion with dynamic droplet volume variation (Princeton U.).
2. Nonlinear forecasting analysis of complex spatiotemporal behaviour in SES (Tokyo U. of Science).

This work builds upon the extensive expertise of the PI on accurate state-of-the-art modelling, theoretical analysis and computations, and is both intradisciplinary (in linking together areas within chem. eng.), and interdisciplinary, in connecting the fields of applied maths, physics and chem. eng. Also, the PI was already collaborating with JBB-YGK and HG. However, the proposed OTG greatly contributed towards cementing and enhancing these collaborations. It also helped SK to expose himself to new concepts and methodologies, e.g. the "equation-free" computational framework, developed in the group of YGK, that allows a systematic connection between macroscopic, continuum numerical analysis and microscopic-atomistic-stochastic simulators. Or the use of sophisticated complex networks, such as cycle networks and phase space networks, a central theme of HG's research, aiming to uncover hidden features in nonlinear phenomena, e.g. high-dimensionality, scale-free and small world nature. Furthermore, the OTG facilitated the formulation of novel
concepts and ideas for future research proposals and further collaborations.
Exploitation Route The impact plan the PI has developed is based on the identification of several roots to societal-economic impact and impact on peers.

The following activities offer a means of dissemination of the results obtained in this proposal to fellow academics as well as industrial researchers: (i) Our national and international network of academic contacts; (ii) National and international conferences. These are leading international forums for the exchange of information on all aspects of applied mathematics, fluid flow and thermodynamics-statistical mechanics of fluids; (iii) The PI has strong records in publication of articles in the leading international journals of high impact factor (hence read by a wide audience) of our respective fields. Some of the results from this project have already been submitted to high-profile international journals devoted to the publication of authoritative articles at the forefront of the topics of the proposed research; (iv) Research seminars in the UK and overseas; (v) Sabbatical and occasional visits by overseas scientists to the UK and visits to overseas
institutions. We have active international collaborations and regularly visit overseas institutions and receive visitors
from abroad. This will aid to the dissemination of the research and increase its impact; (vi) Web-based dissemination
through the PI's group website; (vii) As noted below there are linkages with industry, through other research.

This is fundamental research, however the applications are far reaching and have potential for commercial exploitation. IC has a dedicated commercial exploitation group, Imperial Innovations Ltd (IIL), and consultancy arm while the engineering faculty has a Knowledge Transfer Fellow and Intellectual Property is protected through IIL. There are also links with industry through other research and, therefore, we are ideally placed to disseminate the relevant parts of this research to those who could ultimately use it in applications. These industrial collaborations are vital resources that will be used to facilitate the wider impact of the research and to ensure that results are channelled towards appropriate applications.

Theme 1 of the OTG in particular, focuses on microscale interfacial flows which play a crucial role in key rapidly developing technologies in the UK's biomedical, smart materials, pharmaceutical and printing industries, to name but a few. Future developments in these sectors will require sophisticated mathematical and computational techniques, as developed in the OTG, which can generate valuable insight into many fundamental aspects of these sectors, and help with optimal product design all the way to production. These are increasingly competitive and knowledge-intensive sectors offering long term research and engineering opportunities. Characterising and understanding the interplay between fl
uid flow and surfaces, one of the main aims of the project, are amongst the basic research questions that must be addressed. This, in the long run, should contribute to the UK's capability to develop integrated multiscale models describing microscale flows, with obvious economic benefits, when considering that the global microfluidic device market is expected to grow to more than $5 Billion. The proposed research will also greatly contribute towards maintaining and enhancing the specialist knowledge of the UK force in multi-scale fluid dynamics and microfluidics.

The applications of Theme 2 of the OTG, which focuses on model noisy weakly nonlinear SPDEs, might appear less obvious, but most physical systems are subject to external or internal random fluctuations with examples ranging from biology and climate modeling to technological applications while synchronization phenomena are abundant in science, nature, engineering, and even social life. Hence, the precise characterization of the influence of noise and synchronization on the long-time dynamics is crucial for the understanding and description of the emerging complex dynamics in several physical, biological and technological settings. Many of these settings can be described by model SES, like the ones identified in the OTG. Harnessing noise and synchronization effects can facilitate the control and optimisation of technological processes, such as the process of nanostructuring of solid surfaces by ion-beam erosion crucial in applied condensed matter research where the "holy grail" is smart surfaces for advanced materials.
Sectors Chemicals,Energy,Environment,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology

URL http://www.imperial.ac.uk/complex-multiphase-systems
 
Description The project was centred on two themes: 1. Dynamics of microscale interfacial flows. 2. Nonlinear forecasting analysis of complex spatiotemporal behaviour in spatially extended systems. In the long term, the research in Theme 1 aims to provide both physical insight in microscale multiphase flows and wetting phenomena initially in open and ultimately in confined and complex geometries and the appropriate modelling/predictive tools for such systems. These should be of great interest to researchers and engineers whose technological processes involve at some stage microscale fluid flows in complex geometries as they will enable them to tackle classes of problems that have hitherto been inaccessible to them. Also to workers in the commercial/private sector with interests and/or stakes in the development of predictive models for systems utilising microscale flows. In terms of applications, these are quite extensive from small-scale separations to microfluidics where, a central problem is how to control and manipulate fluids at small scales, crucial for a variety of applications. The research in Theme 2 appears less obvious in terms of impact and is mostly exploratory, but understanding the effects of noise and synchronization could potentially be crucial in several technological applications, such as nanostructuring of solid surfaces, but also in a wide spectrum of problems from biology and climate modelling to social dynamics. The research facilitated by this project will also lead, in the long term, to the development of state-of-the-art numerical methodologies with the capability of providing accurate and reliable multiscale (from the molecular to the macroscale) simulations of three-dimensional multiphase flows in confined geometry. Such codes do not exist at present. These computational tools will be of benefit to the control and optimisation of microscale industrial processes and devices that exploit microscale flows as they would allow their rapid design and also to designer surfaces for targeted microfluidic applications. High-quality software is a pre-requisite to economic impact and invaluable platform to interact with end users, even at the basic research stage.
First Year Of Impact 2015
Sector Chemicals,Energy,Environment,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology
 
Title Nonlinear dynamics of microscale interfacial flows and model nonlinear partial differential equations 
Description As part of this project, we developed novel methodologies for microscale interfacial flows-microfluidic systems but also generic forecasting computational tools for spatially extended systems (SES). 
Type Of Material Improvements to research infrastructure 
Provided To Others? No  
Impact --We analysed droplet motion on chemically or topographically disordered substrates, and, in particular we characterised hysteresis effects due to the presence of disorder. --We developed nonlinear forecasting methodologies for SES. --We scrutinised the effects of noise on SES, with particular emphasis on distinguishing between noise and stochasticity. We further analysed synchronisation in noisy SES. Microscale interfacial flows which play a crucial role in key rapidly developing technologies in the UK's biomedical, smart materials, pharmaceutical and printing industries, to name but a few. Future developments in these sectors will require sophisticated mathematical and computational techniques, as proposed here, which can generate valuable insight into many aspects, and help with optimal product design all the way to production. These are increasingly competitive and knowledge-intensive sectors offering long term research and engineering opportunities. Characterising and understanding the interplay between fluid flow and surfaces, one of the main aims of the project, are amongst the basic research questions that must be addressed. This, in the long run, should contribute to the UK's capability to develop integrated multiscale models describing microscale flows, with obvious economic benefits, when considering that the global microfluidic device market is expected to grow to more than $5 Billion. The research also greatly contributed towards maintaining and enhancing the specialist knowledge of the UK force in multi-scale fluid dynamics and microfluidics. The applications of the second theme in the project, nonlinear dynamics of SES, might appear less obvious, but most physical systems are subject to external or internal random fluctuations with examples ranging from biology and climate modelling to technological applications while synchronization phenomena are abundant in science, nature, engineering, and even social life. Hence, the precise characterization of the influence of noise and synchronization on the long-time dynamics is crucial for the understanding and description of the emerging complex dynamics in several physical, biological and technological settings. Many of these settings can be described by model SES, like the ones identified here. Harnessing noise and synchronization effects can facilitate the control and optimisation of technological processes, such as the process of nanostructuring of solid surfaces by ion-beam erosion, crucial in applied condensed matter research where the "holy grail" is smart surfaces for advanced materials. 
URL http://www.imperial.ac.uk/complex-multiscale-systems
 
Title Deevlopment of new theoretical and computational tools for microscale interfacial flows and spatially extended systems 
Description The project focused on two main themes each of which involves back-and-forth between theory and computations, fundamental and applied aspects: --Droplet motion with dynamic droplet volume variation (the work was undertaken at Princeton University). --Droplet motion with dynamic droplet volume variation (the work was undertaken at the Tokyo University of Science). 
Type Of Material Computer model/algorithm 
Provided To Others? No  
Impact The first theme focuses on microscale interfacial flows. Such flows underpin a number of rapidly developing technologies in the UK's biomedical, smart materials, pharmaceutical and printing industries, to name but a few. Future developments in these sectors will require sophisticated mathematical and computational techniques which can generate valuable insight into many aspects, and help with optimal product design all the way to production. These are increasingly competitive and knowledge-intensive sectors offering long term research and engineering opportunities. Characterising and understanding the interplay between fluid flow and surfaces, one of the main aims of the project, are amongst the basic research questions that must be addressed. This, in the long run, should contribute to the UK's capability to develop integrated multiscale models describing microscale flows, with obvious economic benefits. The research also greatly contributed towards maintaining and enhancing the specialist knowledge of the UK force in multi-scale fluid dynamics and microfluidics. The applications of the second theme which deals with model weakly nonlinear SPDEs, might appear less obvious, but as already emphasized, most physical systems are subject to external or internal random fluctuations with examples ranging from biology and climate modelling to technological applications while synchronization phenomena are abundant in science, nature, engineering, and even social life. Hence, the precise characterization of the influence of noise and synchronization on the long-time dynamics is crucial for the understanding and description of the emerging complex dynamics in several physical, biological and technological settings. Many of these settings can be described by model SES, like the ones identified here. Harnessing noise and synchronization effects can facilitate the control and optimisation of several technological processes, such as the process of nanostructuring of solid surfaces by ion-beam erosion, crucial in applied condensed matter research where the "holy grail" is smart surfaces for advanced materials. 
URL http://www.imperial.ac.uk/complex-multiscale-systems
 
Description Collaboration with Prof. H.A. Stone 
Organisation Princeton University
Department Department of Mechanical and Aerospace Engineering
Country United States 
Sector Academic/University 
PI Contribution The particular collaborative project with Prof. Stone that resulted from this grant, that of healing of capillary films, involves continuation and bifurcation analysis of self-similar problems. In my group we have extensive expertise on applied dynamical systems, including continuation and bifurcation analysis.
Collaborator Contribution Formulation and physics of the problem.
Impact We have submitted one paper for publication with Prof. Stone as co-author. Also, extensive discussions with Prof. Stone led to byproducts-further studies with two additional manuscripts submitted for publication.
Start Year 2015
 
Description Collaboration with Prof. J.B. Benziger 
Organisation Princeton University
Department Department of Chemical and Biological Engineering
Country United States 
Sector Academic/University 
PI Contribution The joint project with Prof. Benziger involves characterising complex multiphase flows in confined and structured geometry. In my group we have developed an arsenal of tools based on diffuse interface/Cahn-Hilliard models to tackle multiphase flows.
Collaborator Contribution Prof. Benziger and his team have done extensive experiments on complex multiphase flows in confined and structured geometry. The aim is to combine our theoretical-computational results with the experimental ones and also use the theory-computations to guide the experiments, i.e. suggest intelligent experiments in certain regions of the parameter space which might not be obvious from the outset to experimentalists.
Impact Fluid transport is crucial in a wide spectrum of technological applications involving multiphase flows often in complex geometries, including porous membranes, microstructured channels or micropatterned surfaces. Examples are found in many areas, from microchemical and biological engineering to materials processing and the rapidly growing fields of micro- and nanouidics. An important application of current interest is that of polymer electrolyte membrane (PEM) fuel cells, of particular interest to Prof. Benziger and his team.
Start Year 2015
 
Description Collaboration with Prof. M.A. Fontelos 
Organisation Institute of Mathematical Sciences
Country India 
Sector Academic/University 
PI Contribution The particular project here was a byproduct of extensive discussions with Prof. Stone on thin-film rupture. And the collaboration in fact with Prof. Fontelos was initiated by Prof. Stone. The project aims to understand the mathematical details of singularity formation in thin-film flows, in particular flows where the film ruptures in finite time. My team has expertise in asymptotic analysis but also applied dynamical systems including continuation and bifurcation analysis, all crucial for the successful undertaken of the project.
Collaborator Contribution Prof. Fontelos' expertise is on asymptotic and self-similar analysis of nonlinear partial differential equations, crucial for the project.
Impact So far we have submitted three papers with Prof. Fontelos. Our collaboration continues, and one of the group members in my group, Dr. Michael Dallaston, will be visiting Prof. Fontelos in Madrid for a week over the summer of 2017.
Start Year 2015
 
Description Collaboration with Prof. Y. Kevrekidis 
Organisation Princeton University
Department Department of Chemical and Biological Engineering
Country United States 
Sector Academic/University 
PI Contribution The three main projects resulted from this collaboration are: --Fluid motion in complex geometry, e.g. droplets on chemical heterogeneous substrates and multiphase flows in confinement. My group has extensive expertise in tackling fluid flow problems in different settings. --"Equation-free" approach to the Navier-Stokes (NS) like equation we have obtained in my group from our dynamic density-functional theory formalism (DDFT) we have developed in my group. --Time-reversibility in molecular dynamics (MD). In my group we have recently branched out on MD and related molecular simulations for fluids.
Collaborator Contribution Prof. Kevrekidis expertise is in the general area of applied and computational mathematics, in particular applied dynamical systems and dimension reduction methodologies, all crucial for the successful undertaking of the above projects. But also, Prof. Kevrekidis has developed the so-called "equation-free" computational framework that allows for a systematic connection between macroscopic descriptions, continuum numerical scheme and microscopic-atomistic-stochastic simulators, crucial for the second of the projects above where we are trying to benchmark the unknown viscosity coefficients in our NS-like models from MD.
Impact So far we've had one paper published and we hope for additional papers once we make progress on the above projects.
Start Year 2015