Viscoelastic Flows of White-Metzner Type Fluids

Polymer melts and solutions are highly important materials in industry, and display characteristics of both viscous fluids and elastic solids. Due to this, these viscoelastic materials can be quite hard to model effectively, and require a constitutive relation in order to relate any material deformation to the internal stresses. This is particularly the case in many polymer processing applications, where these materials are encountered in situations such as flow round a re-entrant corner, and flow out of a die. These geometries are linked by the existence of stress singularities, and failure to understand the mathematics of these singularities can ultimately lead to negative implications on the quality of any processed products.

Most of the early work on understanding this viscoelastic flow was completed by chemical engineers, who took a largely data-driven approach to proposing and testing models for these materials; analytical approaches were rarely attempted. However, in the last thirty years, this has begun to change via the usage of asymptotic analysis, which can produce a simpler 'leading order' description of the flow whilst retaining the key underlying physics.

The main aim of this project is to build on the success of these techniques by applying them to classes of viscoelastic fluids governed by White-Metzner type constitutive relations. In particular, flow around a re-entrant corner will be studied, in which leading order governing equations will be constructed via the use of boundary layer theory, and by utilising a natural stress formulation to describe the stresses within the fluid. These equations will then be analysed numerically, and their suitability for describing the re-entrant corner flow will be assessed via comparisons with simulations from computational fluid dynamics (CFD) software such as OpenFOAM.

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.