Spatially localized states in the Zhang-Vinals equation

Spatially localized states states are typically comprised of a pattern in a well-bounded region in space and surrounded by a homogeneous background. These are well understood in the stationary regime but very little is known about their time-dependent counterparts. We are here interested in oscillons, which are spatially localized states in which the pattern is oscillating. A striking example of this is found in Lioubashevski et al., Phys. Rev. Lett. 83, 3190--3193 (1999). In this paper, a complex fluid is vibrated vertically and displays oscillating peaks in the midst of a non-oscillating background. There is no theoretical work available on the fluid equations that could shed light on these structures but fortunately, the Ginzburg--Landau equation, known to model the dynamics close to onset with great accuracy, is giving precious guidance for upcoming work (see A.S. Alnahdi, J. Niesen and A.M. Rucklidge. SIAM J. Applied Dynamical Systems 13 (2014) 1311-1327).
The idea for this project is to consider a model of a vibrating fluid layer known as the Zhang--Vinals model and, guided by the work on the Ginzburg--Landau equation above, find and analyse the formation of these oscillons. The work would focus mainly on the nonlinear aspects of the configuration (pattern formation), with some asymptotics (derivation of the Ginzburg--Landau equation), analytical methods (derivation of the Zhang--Vinals model) and some numerics (integrating the Zhang--Vinals model).

The impact and benefits will reach multiple stakeholders.

(i) CDT Students:- Will develop substantial technical and transferable skills enabling them to build a career and become leaders in industry or academia. In addition to a wide range of computational, modelling and experimental techniques, students will have many opportunities to develop team working, communication and problem solving skills. Students will have very strong career prospects with a wide range of options, including industry and public sector.

(ii) End-user partners:- Will gain access to a pool of at least 50 skilled graduates to innovate in their business and to realise direct impact from research outcomes: new products, processes, and tools. New or strengthened collaborations with academic partners will also follow.

(iii) Academic overseas collaborators:- will share new research outputs, stronger partnerships with Leeds, and knowledge exchange on tools and techniques: thus benefiting research outcomes and researcher training in both countries.

(iv) Other students:- Will have the opportunity to visit Leeds, whilst future students will have access to the new tools and techniques developed by the CDT for learning, thus inspiring new UG/MSc/PhD projects.

(v) Research at Leeds:- We will consolidate our critical mass of fluids-based research through the development of a "cohort of academics", as well as cohorts of students. New research outputs and new collaborations (across Leeds, with industry and overseas) will follow, and we will promote our large body of work coherently with external partners and to the media.

(vi) Other industry:- New tools, processes and techniques developed through research during the CDT will be disseminated via industrial as well as academic routes. We will pro-actively encourage new partners to engage with the CDT as it evolves.

(vii) The economy:- Skilled graduates are key to economic growth and ours will contribute to challenge areas such as energy, the environment, the health sector, as well as those with chronic skills shortage such as the nuclear industry. Innovation, typically in partnership with industry, will lead to economic benefits such as new products, services and spin out.

(viii) Society:- Research leading to new insights into energy, the environment and health challenges will lead to healthier, safer and more efficient environments for the public. Public engagement activities will raise the profile of Fluid Dynamics, and enable the public to understand its enormous breadth of application, and importance, to real world problems.

Evidence for impact creation comes partly from government sponsored reports pointing to the need for well-trained graduates in fluid dynamics, and also from the many letters of support we have received from our partners. In consumer products P&G tell us that "within our current product portfolio, fluids feature in 21 of our 24 one billion dollar brands (more than $1 billion sales) which include detergents, shampoos, fabric softener, dishwashing liquid, batteries, toothpaste and cosmetics". In engineering design Parker Hannifin believe that "the UK will need a greater number of graduates with complementary skills in high fidelity CFD and optimisation methods". There is a similar demand in the environmental sector. For example the National Oceanography Centre state that "in the coming years we expect to build our technical expertise in areas such as numerical methods, unstructured gridding and solvers, ocean dynamics, buoyancy driven flows and ensemble methods for uncertainty estimates", while HR Wallingford "expect to require access to expertise in relevant physical processes, compressible/incompressible flow, physical model scaling, numerical methods, multi-phase flow, atmospheric flows".