Industrial mathematics

Research staff: Chris Budd, Jonathan Dawes, Matthias Ehrhardt, Jonathan Evans, Melina Freitag, Ivan Graham, Apala Majumdar, Eike Müller, Rob Scheichl, Alastair Spence

Industry provides a wealth of stimulating and challenging quantitative scientific problems. Many of these can be tackled using applied mathematics and numerical computation; this leads not just to satisfying applications of the mathematical ideas but often also provokes new questions and lines of research.

Examples of recent research collaborations include work on problems as diverse as freezing fish, safety problems in nuclear reactors, corrosion of metals, digital communications, 'squealing' in railway bogies, coal burning, and food digestion.

Staff in Mathematical Sciences regularly take part in European Study Group with Industry meetings (and Bath has hosted a Study Group in the past) and work with industry in a variety of ways: for example the department has an excellent record in facilitating Industrial CASE PhD studentships in which research students spend periods of time seconded to an industrial partner who is an enthusiastic supporter of the research programme undertaken for the PhD.

Within the University, we maintain close links with the Departments of Mechanical, and Electrical & Electronic Engineering. These links include joint projects and jointly supervised research students.

Mathematical biology

Research staff: Ben Adams, Ben Ashby, Nick Britton, Dorothy Buck, Paul Milewski, Tim Rogers, Hartmut Schwetlick, Jane White, Kit Yates

Mathematical biology is concerned with the quantitative modelling of biological processes, at scales from individual cells up to entire ecosystems comprising populations of many different species.

Biological systems are formidably complex. They operate on multiple scales, from the molecule up to the ecosystem, and they are inhomogeneous at each level. For example, many strains of a typical contagious disease arise through mutation and circulate simultaneously, infecting a population composed of individuals of different ages and immune histories moving in space and interacting through chance encounters and a changing social network.

It has long been recognised that the analysis and computational solution of simple homogeneous deterministic mathematical models at a single level often helps develop insight into a biological system. However, in modern mathematical biology it is considered increasingly important to take account of the complexity of the system, exploiting its structure to allow progress in understanding its behaviour. Mathematical biologists must therefore be excellent mathematical modellers; they must have a deep understanding of the biological system, gained from the literature and through close collaboration with biologists; they must also keep abreast of developments in diverse areas of mathematics that may allow them to analyse ever more complex models as effectively as possible.

Mathematical biology at Bath is a truly interdisciplinary subject. The mathematical biologists in the department are actively engaged in research with colleagues from various other disciplines in the university, and beyond. Within the university a close relationship between the disciplines is maintained by the Centre for Mathematical Biology (CMB), a group of mathematicians, biologists and other scientists with interests in the mathematical modelling and analysis of biological systems, who maintain close collaboration and hold frequent meetings.

Continuum mechanics of solids and fluids

Continuum mechanics is concerned with describing the deformation of solids and the flow of liquids and gases. The dynamics of the Earth's atmosphere and oceans is a rich source of mathematical problems. Many problems concern wave motion and the transport of energy: the generation and absorption of waves is central to our understanding of many geophysical phenomena, from under-sea earthquakes to fluctuations in the jet stream in the lower stratosphere.

In addition to geophysical phenomena, our research includes work on problems that lie between traditional fluids and solids, such as viscoelastic fluids, granular materials, porous flows and composites.

Research in continuum mechanics and waves at Bath has broadened considerably in recent years through a number of new appointments and the development of strong links with industry, including in particular the UK Meteorological Office. Weekly seminars are held, divided into a formal series with external speakers, and an informal series which both research students and staff contribute to. Regular visitors, with long-standing connections to the Department, include Valery Smyshlyaev and John Willis FRS.

The Centre for Nonlinear Mechanics provides a convenient unifying framework for forming research links with other departments in Bath; notably Mechanical Engineering and Electrical and Electronic Engineering.

Solid mechanics

Kinetic relations for phase-transforming materials

Research staff: Johannes Zimmer

Phase transitions in solids can often be described in the framework of evolution equations in elasticity. One particular area of interest to us is the study of a moving interface. Kinetic relations state the dependence of the interface's velocity on the configurational force. A number of successful models in engineering postulate a phenomenological kinetic relation to make the equations of motion on the continuum level well-posed. Our aim is to derive kinetic relations rigorously in the context of the passage from the atomic level to the continuum model.

Variational problems in nonlinear elasticity

Research staff: Jey Sivaloganathan

This research covers a wide spectrum of variational problems arising in nonlinear elasticity. Particular problems of current interest include the existence and properties of singular minimizers and variational models for fracture and defect/crack initiation, cavitation problems with further links to liquid crystals and cavitation in fluids, conservation laws in weak form and symmetries in variational problems, numerical methods to detect singular minimisers.

Singular structures

Research staff: Roger Moser

The properties of materials are often determined to a considerable extent by singular structures. In continuum models based on variational principles or PDEs, these structures often arise as features of the limits for an asymptotic analysis. We aim to derive the form of the singular structures and equations for their behaviour rigorously from the underlying models. The problems studied include Ginzburg-Landau vortices in various contexts, domain walls in ferromagnets, and crystal facets.

Fluid mechanics

Buoyancy-driven instabilities and pattern formation

Research staff: Jonathan Dawes

Many viscous fluid flows are produced by external forces that act on the whole body of fluid, for example gravity. Such external forces drive fluid flow through buoyancy effects due to gradients in temperature or solute concentrations. Often these flows show a spontaneous structure that has features that are experimentally observed not to depend strongly on boundary conditions. Moreover, such flows often undergo qualitative changes (bifurcations) at critical values of control parameters. The investigation of these qualitative changes, and the spontaneous emergence of spatial structure in flow fluids, provides a rich source of mathematical problems, many involving the application of ideas from dynamical systems theory.

Complex fluids

Research staff: Jonathan Evans

The complex fluids of interest are those possessing both solid and liquid like behaviour (i.e. viscoelastic fluids). These fluids have memory which leads to new and intriguing behaviours whilst on the other hand introducing tremendous complications for their mathematical study. Nonlinear differential models are considered which typically describe polymeric fluids such as plastics, oils, paints, biological fluids and foods to name but a few instances of their occurences. Analysis of such models is still in its infancy relative to Newtonian fluids and many important problems remain unsolved. Active research is involved with the use of asymptotic methods to derive approximate solutions in commonly occurring geometries e.g. flow around sharp corners, wedge flows, free surface flows. Technical tools involve matched asymptotic expansions, similarity methods as well as supporting numerical analysis. These tools allow, for example, boundary layer and stability analysis.

Fluid mechanics continued

Liquid crystals

Research staff: Apala Majumdar

Geophysical fluid mechanics and waves

Research staff: Chris Budd, Jonathan Dawes, Paul Milewski

Geophysical fluid mechanics is a broad research field encompassing spatial scales from wind ripples to planetary circulations and timescales from seconds to climate studies. Our group has expertise in the mathematical study of surface and internal waves and their impact. Surface waves are spectacular, sometimes dangerous and destructive such as in the case of rogue waves and tsunamis, and also are a crucial part of atmosphere-ocean interactions. Although their mathematical study is over a century old, our understanding of more complex and nonlinear phenomena is limited. Analytical and computational issues are extremely challenging, particularly because the boundary conditions are both nonlinear and the boundary motion (the waves themselves) is a priori unknown. Density stratified fluids such as the ocean and atmosphere also support propagating waves. These waves play an important role in energy propagation, dissipation and fluid mixing and are ultimately a crucial ingredient in understanding climate. They are often subject to, and generate their own shear, leading to mathematical issues of stability and well posedness. Breaking internal waves mix the fluid and change the environment upon which they propagate: an extremely challenging problem to model mathematically.