Coherent structures in non-local active-dissipative equations: theory and computations

The aim of the proposed research is to analyse coherent structures for non-local active-dissipative partial differential equations (PDEs). Such equations are characterised by the presence of mechanisms of instability (energy production) and stability (energy dissipation). In addition, such equations contain non-local terms that cannot be expressed as polynomials or functions of the solution to be found and its derivatives. The solutions of active-dissipative equations are often characterised by a particular spatial and temporal behaviour, e.g. by space, time or space-time localised structures, the so-called coherent structures. The study of such structures has received a lot of attention over recent years due to their importance in hydrodynamics, nonlinear optics, chemical-reaction theory and mathematical biology. Despite recent developments in the theory of such structures for 'local' equations, coherent-structures interaction in non-local active-dissipative equations has not been developed as of yet and new mathematical techniques need to be introduced. The proposed research is a decisive step towards the understanding of the formation and interaction of coherent structures for such non-local equations with the ultimate goal to advance the understanding of non-local effects in PDEs, which is important both from the theoretical point of view and for numerous practical applications. The research is particularly timely because of its relevance to a wide range of industrial devices that utilise thin liquid films in the presence of various external effects and complexities. Examples include cooling systems, falling-film chemical reactors, compact reflux condensers and microfluidic devices.

It is anticipated that the impact of the proposed research will be both academic and non-academic at the highest international level. Besides the immediate beneficiaries including applied mathematicians and physicists in the UK and overseas working in the fields of infinite-dimensional dynamical systems, interfacial fluid mechanics and thin-film flows, beneficiaries will also include geophysicists and biologists. There is a wide variety of natural phenomena (e.g. propagation of internal waves in a deep stratified fluid, certain types of continental-shelf waves, population dynamics, neuronal networks) where non-local equations arise. Therefore, the proposed work will improve understanding of the various processes and phenomena described by such equations.

The research will also be of interest to a wide community of experimentalists, engineers and workers in the commercial private sector, whose technological processes involve thin liquid films in the presence of various external effects and complexities such as an applied electric field or a turbulent gas flowing above the liquid film. In terms of practical applications, there is a broad range of technological processes and industries where liquid films subjected to various external effects are key components. This includes coating processes, cooling systems, falling-film chemical reactors used in the production of detergents, compact reflux condensers, desiccant cooling systems and also microfluidic devices. The spatio-temporal dynamics of the interface in these applications is often characterised by continuously interacting nonlinear wave patterns (coherent structures) that are known to profoundly affect the heat and mass transfer in industrial units. The equations that arise in the study of the dynamics of the interface in such applications are often non-local. The proposed research will improve theoretical understanding of such equations and will therefore enable to optimise processes utilising liquid films. A successful conclusion to the proposed work will therefore be of benefit for various industries. This will in turn bring obvious benefits for the wider public, society and economy.

In addition, the proposed research will have an educational impact. First, it will be an excellent training opportunity for the PDRA. Second, simplified versions of some parts of the proposed research will be given in the future to interested undergraduate students as part of the undergraduate project module at Loughborough University. This will provide the students with a research experience and with skills and confidence needed to proceed with postgraduate studies in the fields of applied and computational mathematics. This will therefore result in increasing the specialist knowledge of the UK force in these areas.