Elucidating the lubrication layer arising from impacting droplets

When a small droplet impacts on a bath of the same fluid it usually coalesces. In certain regimes, however, a thin layer of air remains between the droplet and the bath acting as a barrier, and as such it is possible for the droplet to be propelled upwards due to the restorative capillary force of the bath, before this air layer is depleted. If the bath is vibrated at a suitable vibration frequency, this rebounding mechanism can repeat periodically causing the droplet to bounce indefinitely off the surface of the bath. In addition, for a vibrating bath, Faraday waves are generated radiating outwards from the impact site on the surface of the bath. As the vibration amplitude is increased, the rebounds are higher and the resulting Faraday waves increase in amplitude, eventually destabilising the droplet causing it to 'walk' across the surface of the bath. These phenomena provide motivation for the problem at hand, as the main topic of study will be the role of the lubrication air layer that allows for this droplet, and also similar solid-fluid impacts at the capillary scale, to exhibit rebound.

The first goal will be to determine the air layer dynamics. This will need to be considered first for a solid sphere impact. It is expected that this will require several asymptotic regimes: the initial impact when the lubrication effect begins to dominate; the intermediate stage when the layer is well developed and expanding or contracting around the sphere; and the final stage of deconnection and liftoff. The main area of fluid dynamics research that will be used is lubrication theory, which is applicable when the thickness of the fluid being considered is significantly smaller than the horizontal lengthscale of the fluid. This allows for simplifying assumptions to be made that reduce the problem to something more computationally inexpensive than current attempts at numerically solving the full Navier-Stokes equations. Following the solid impact problem we will consider liquid drop impacts. Some more speculative questions that could be addressed in this part of the research are whether there is an effective "contact angle" for this problem that can be rationalised, and whether vorticity can be generated in the fluid bath due to stresses at the effective contact line.

We will also consider how the pressure created in the air layer feeds into the Faraday wave generation. This follows on from "kinematic match models" (Galeano-Rios, Milewski and Vanden-Broeck) which ignore the air layer altogether, but achieve Faraday wave generation by evolving a "pressed area". We then aim to combine the two main directions of work into one cohesive joint model for the drop and the waves. For the second stream of research we will be utilising quasi-potential fluid dynamics to describe the motion of the waves. The project will involve carefully designed numerical methods to simulate the evolution of the resulting PDEs in Matlab.

The supervisory team will be led by Prof Paul Milewski (50%), with Dr Jonathan Evans (25%) and Dr Phil Trinh (25%) as secondary supervisors. Drs Evans and Trinh bring expertise in asymptotic methods and lubrication theory. It is also expected that external collaborations with Dr. Radu Cimpeanu (Warwick) and Prof. Daniel Harris (Brown) who have expertise in CFD simulations and experimental measurements for this problem.

Combining specialised modelling techniques with complex data analysis in order to deliver prediction with quantified uncertainties lies at the heart of many of the major challenges facing UK industry and society over the next decades. Indeed, the recent Government Office for Science report "Computational Modelling, Technological Futures, 2018" specifies putting the UK at the forefront of the data revolution as one of their Grand Challenges.

The beneficiaries of our research portfolio will include a wide range of UK industrial sectors such as the pharmaceutical industry, risk consultancy, telecommunications and advanced materials, as well as government bodies, including the NHS, the Met Office and the Environment Agency.

Examples of current impactful projects pursued by students and in collaboration with stake-holders include:

- Using machine learning techniques to develop automated assessment of psoriatic arthritis from hand X-Rays, freeing up consultants' time (with the NHS).

- Uncertainty quantification for the Neutron Transport Equation improving nuclear reactor safety (co-funded by Wood).

- Optimising the resilience and self-configuration of communication networks with the help of random graph colouring problems (co-funded by BT).

- Risk quantification of failure cascades on oil platforms by using Bayesian networks to improve safety assessment for certification (co-funded by DNV-GL).

- Krylov regularisation in a Bayesian framework for low-resolution Nuclear Magnetic Resonance to assess properties of porous media for real-time exploration (co-funded by Schlumberger).

- Machine learning methods to untangle oceanographic sound data for a variety of goals in including the protection of wildlife in shipping lanes (with the Department of Physics).

Future committed partners for SAMBa 2.0 are: BT, Syngenta, Schlumberger, DNV GL, Wood, ONS, AstraZeneca, Roche, Diamond Light Source, GKN, NHS, NPL, Environment Agency, Novartis, Cytel, Mango, Moogsoft, Willis Towers Watson.

SAMBa's core mission is to train the next generation of academic and industrial researchers with the breadth and depth of skills necessary to address these challenges. SAMBa's most sustained impact will be through the contributions these researchers make over the longer term of their careers. To set the students up with the skills needed to maximise this impact, SAMBa has developed a bespoke training experience in collaboration with industry, at the heart of its activities. Integrative Think Tanks (ITTs) are week-long workshops in which industrial partners present high-level research challenges to students and academics. All participants work collaboratively to formulate mathematical
models and questions that address the challenges. These outputs are meaningful both to the non-academic partner, and as a mechanism for identifying mathematical topics which are suitable for PhD research. Through the co-ownership of collaboratively developed projects, SAMBa has the capacity to lead industry in capitalising on recent advances in mathematics. ITTs occur twice a year and excel in the process of problem distillation and formulation, resulting in an exemplary environment for developing impactful projects.

SAMBa's impact on the student experience will be profound, with training in a broad range of mathematical areas, in team working, in academic-industrial collaborations, and in developing skills in communicating with specialist and generalist audiences about their research. Experience with current SAMBa students has proven that these skills are highly prized: "The SAMBa approach was a great template for setting up a productive, creative and collaborative atmosphere. The commitment of the students in getting involved with unfamiliar areas of research and applying their experience towards producing solutions was very impressive." - Dr Mike Marsh, Space weather researcher, Met Office.