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Nonlinear, but under control: a hierarchical modelling approach to manipulating liquid films

Supervisors: Radu Cimpeanu, Susana N. Gomes

We are surrounded by situations that depend on a controlled outcome in our day-to-day lives, ranging from controlling the evacuation of crowds, to efficient drug delivery, or cooling systems inside computing centres. Most real-life scenarios rely on complicated models which are too complex to tackle analytically or computationally. Using the framework provided by a beautiful and rich physical problem – controlling nonlinear waves in falling liquid films – the project will provide opportunities to develop analytical and computational multi-physics tools. Acting in tandem for the first time, they become sufficiently powerful to translate robust theoretical strategies into realistic technological solutions.

Whilst fundamental in nature, the project brings together elements from diverse areas in modern applied mathematics and it aims to provide a new framework for hierarchical modelling in control theory for complex physical systems that is intrinsically interdisciplinary. The methodology to be developed can be summarised as:

1. Weakly nonlinear models: the simpler models we consider, highly versatile and efficient. They provide an environment where mathematical analysis, control design and rapid numerical calculations are possible. However they are only applicable for very simple scenarios.

2. Advanced reduced-order models: more realistic models, but their complex nature renders analytical results almost impossible. However control development based on certain assumptions is still viable, and the resulting controls were recently shown to be reliable.

3. Direct numerical simulations: provide highly accurate solutions of the full model and do not rely on any modelling assumptions. However they are restrictively expensive unless efficiently guided.  

The project will thrive at the interface between modelling, novel analytical approaches and high performance computing, being naturally linked to several applications in lens and microchip manufacturing, advanced coating systems and next-generation cooling technologies.