Feedback Control of Deformable Bubbles

Many physical objects can, in principle, balance perfectly in a number of different orientations, but only some of these orientations are stable. For example, if we try to balance a bicycle upright, it will always fall to one side or another - we say that this is an unstable steady state. We can prevent the bicycle from falling over by continually observing the system and making frequent adjustments based on the latest observations. These adjustments and observations are called feedback control, and if chosen carefully will enable the system to remain in unstable steady states.

In this research project, we will use feedback control in the form of "control-based continuation" to investigate the unstable steady states of a deformable air bubble moving through a very viscous fluid. This is a classical problem in applied mathematics, and has applications in microfluidic devices, oil extraction and airway reopening. Previous experiments and mathematical modelling for this system have shown that the bubble can propagate in a wide range of different shapes, but without feedback control these experiments were unable to definitively identify the unstable possibilities and hence to fully test the validity of the model predictions.

The idea of control-based continuation is to apply feedback control towards a wide range of different target states, and to seek in particular target states where no input is required for the bubble to remain exactly at equilibrium. These states would be steady states even without any control, and are stabilised by the real-time feedback. Importantly, the control-based continuation process can be carried out equally well in an experimental setup or in a numerical simulation. In this project, we will use numerical testing and theoretical analysis to develop control-based continuation routines that can work when subject to realistic constraints, and will carry out simple experiments to show this succeeding in practice.

The propagation of a deformable air bubble is a rich scientific problem and has been studied using different forms of applied mathematics for many years. However, the feedback control techniques we develop here should ultimately be applicable across a much broader range of systems. These techniques would be particularly important for systems where theoretical models are not yet developed or are very complicated (such as blood cells and capsules), but where we would still like to understand and characterise the body motion or interface deformation in fluid flow.

This research proposal is concerned with developing control-based continuation techniques for use in free-surface fluid mechanics for the first time; these techniques will enable new and powerful investigative techniques involving combinations of theory and experiments for use in academic or industrial free-surface fluid and soft matter problems in a range of engineering and physics fields. The emphasis on interdisciplinary approaches will also benefit the project team, and the UK both via skilled scientists and through new outreach and social media activities targeted at specific sectors.

Specifically, we identify the following beneficiaries:

- Industrial research areas encompassing free-surface fluid dynamics include microfluidics, oil recovery, ink-jet printing and curtain coating. Control-based continuation can greatly facilitate the use of experiments and observations in characterising and mapping unstable behaviour. This improved capability will have direct relevance in design, testing and validation of new technologies and devices, especially in cases where theoretical models are not available or not entirely reliable.

- The project team: the principal investigator and postdoctoral research associate working on this project will both benefit from the grant in terms of research independence, career development and technical skills. They will also gain critical experience in developing and pursuing an innovative project that draws strength from the interface between applied mathematics and engineering disciplines.

- The UK will benefit via the provision of skilled and innovative scientists who have experience of working at the interface between fluid dynamics, control engineering and applied mathematics, experience developing new techniques combining theory, experiment and cutting-edge control theory, and in working as part of a multi-disciplinary team.

- Sixth-form students, at a crucial point for making career choices, will benefit from a range of targetted outreach activities emphasising the interaction of different STEM subjects, here via applied mathematics and engineering combining to resolve long-standing problems. Furthermore, we aim to characterise the possible deformations of a bubble, and a range of multi-media resources will be developed to explain this highly visual outcome to a range of audiences.

Impact will be enhanced through two particular features of this grant proposal. Firstly, experiments corresponding to the problems in work packages 1 and 2 of the theoretical programme will be performed by a PhD student funded by the Department of Mathematics at the University of Manchester as part of their support for this grant. Successful demonstration of control in these practical experiments will significantly increase the effectiveness of impact for every beneficiary listed above. Secondly, we will organise and host a workshop on control-based continuation towards the end of this grant, specifically focusing on its relevance to industrial applications and research in fluid dynamics, soft matter and control theory. The University of Manchester has a track record in using such workshops to develop productive ongoing research links. We will build upon our existing strong links with the inkjet printing community, while also working with the business engagement team in Manchester to identify new opportunities for industrial collaborations.