School of Mathematics

Polymer dynamics and rheology
Oliver Harlen, Daniel Read, Mark Kelmanson

Polymers are long molecules made from joining together lots of small molecules (or monomers). Sometimes polymer molecules are linear, but very often - notably in the case of Low Density Polyethylene (LDPE) used to make plastic bottles - they include many branches. During the manufacture of polymeric (or plastic) materials and commodities, liquids containing polymers are subjected to flow. The way these liquids react is determined by the shapes, or configurations that the molecules adopt.

Polymer molecules behave like springs, and become stretched by the flow, giving rise to the strongly elastic behaviour of polymeric fluids. The study of the dynamics of polymer molecules is very important for the understanding of flow of polymeric fluids. If polymer molecules overlap sufficiently, then they get tangled up (like spaghetti) so that they are constrained in their movement. The "tube model" for entangled polymers provides a conceptual framework for understanding the constrained motion, and for making mathematical predictions about the polymers' response to flow. Branchpoints in the polymer molecules provide addtional obstacles to the motion of entangled polymers, so that the distribution of branchpoints in polymer molecules can be a critical factor in determining flow properties.

Inverse problems
Daniel Lesnic

Whilst direct formulations consist of determining the effect of a given cause, in inverse formulations the situation is completely, or partially reversed. The interest is into the research of inverse problems for partial differential equations governing phenomena in fluid flow, elasticity, acoustics, heat transfer, mechanics of aerosols, etc. Typical practical industrial applications relate to flows in porous media, heat conduction in materials, thermal barrier coatings, heat exchangers, corrosion, etc. The objectives are to investigate the existence, uniqueness and stability of the solution to the problem that mathematically models a physical phenomenon under investigation, and to develop new convergent, stable and robust algorithms for obtaining the desired solution. The analyses concern inverse boundary value problems, inverse initial value problems, parameter identification, inverse geometry and source determination problems. Ongoing, the determination of heat transfer coefficients (in heat conductors and polymers) using inverse methods is supported by a prestigious EU-FP7 Marie Curie International Incoming Fellowship.

We are represented on the Academic Advisory Board of the Knowledge Transfer Network for Industrial Mathematics in the UK and for the past 5 years we have organised Industrial Mathematics Inverse Problems Sandpits at which people from industry presented their problems. This resulted in numerous industrial contacts being made, consultancy work being performed, and a couple of EPSRC Industrial Mathematics Cases being awarded.

Non-Newtonian fluids
Oliver Harlen

Elastic stresses in polymeric and other complex fluids can give rise to strange flow behaviour not seen in Newtonian fluids. This can, for example, produce undesirable instabilities in industrial processes. The aim of our research is to predict how these fluids will flow in various flow geometries and to determine the conditions for the flow to become unstable. Our current research includes investigations of how bubbles grow in polyurethane foams; how filaments of polymer stretch and break-up; and flow instabilities in extrusion.

Thin-film industrial (coating) and ophthalmic (tear-film) flows
Mark Kelmanson

Many industrial processes and natural phenomena involve the flow of thin liquid films, perhaps the most widespread industrial application being the deposition on solid surfaces of a thin protective and/or decorative coating. After initial application, the film is usually uneven and hence evolves under competing influences such as gravity, surface tension and inertia. It is frequently required that the final (steady) state of the coating should be a film of uniform thickness, and so it is of practical value to be able to formulate and solve the nonlinear spatio-temporal evolution equations that govern the coating thickness in order to understand and predict the transient dynamics of the film. Our research in this area has led to the development of fully automated computer-algebra procedures for deriving such equations, in which our distinctive automation permits a more complete mathematical modelling of the underlying physics than has hitherto been possible. The equations have moreover been solved to a high (prescribed) degree of accuracy using similarly automated solution methods based on mutiple-timescale asymptotics. As a result, a detailed understanding of how flow evolution explicitly depends upon various physical parameters has been gained. For example, the formation and dissipation of thin-film shocks (in collaboration with Professor John Hinch FRS and Dr Paul Metcalfe, DAMTP) has been expicitly parameterized for roll-coating geometries, in which new limit-cycle phenomena resulting from inertial effects have also been discovered (in collaboration with Dr Christian Groh, Leeds).

Closely related to the above is the study of thin tear films arising in ophthalmic flows, which are complicated by additional effects such as evaporation, a viscoelastic surfactant lipid layer and free-surface dilatation. Our recent research has involved formulating model evolution equations for the tear-film thickness and surfactant concentration that are consistent in their level of approximation, so that the above-mentioned automated procedures can be modified to account for the new geometry and physics. It is hoped that this work (in collaboration with Dr Jonathan Summers and Mr Gareth Hurst, Leeds) will ultimately contribute to a better understanding of the well-known dry-eye phenomenon.

Reaction chemistry and branched polymer architecture
Daniel Read

There are different chemical routes used to produce branched polymers in an industrial setting. The particular reaction chemistry, and the reactor type and conditions, have a large effect on the number and distribution of branches throughout the polymer molecules. Current research is examining this relationship between reaction chemistry and branched polymer architecture with a view towards the reaction design for polymer melts with tailored flow properties.

Polymer dynamics and neutron scattering
Daniel Read

It is important to understand the shapes, or conformations that polymers take under flow conditions. Although polymer rheology (the stress response of the fluid) is one way of probing this, it is important to have other independent tools to check that the theory is right. A more direct measure of polymer shape is obtained via neutron scattering from polymers that are parially labelled with deuterium. We aim to predict neutron scattering patterns from deformed polymer melts.

Suspension mechanics
Oliver Harlen

Another important class of non-Newtonian fluids is suspensions of solid particles, such as spheres and fibres, or droplets or bubbles in Newtonian or non-Newtonian fluids. For example, short glass and carbon fibres are often added to injection moulded plastics to reduce cost and to improve the mechanical or thermal properties of the finished product. Small spheres are often suspended in a fluid in order to transport, for example, pharmaceutical powders around processing plants; and many food-stuffs and skin-care products are formed from oil-in-water emulsions. While the motion of single particles in Newtonian fluids is well understood, suspensions of large numbers of particles that interact through the fluid remain challenging, both analytically and numerically.

Our current research includes investigations of how rough particle surfaces can affect the flow of a suspension of solid spheres; how collisions between fibres affect the flow properties of concentrated fibre suspensions; propagation of sound waves through colloidal suspensions; and how droplets deform in a polymeric fluid flow.

For further information on our research please visit our Group pages.