# Possible PhD Projects in Applied Mathematics

Suggested projects for postgraduate research in Applied Mathematics.

Applied Mathematics research in the School of Mathematics, Statistics and Physics is concentrated in the following areas:

**Cosmology and Quantum Gravity****Astrophysical and Geophysical MHD****Quantum Matter****Mathematical Biology and Archaeology**

If applying for a PhD project in one of these areas, please provide the titles of up to three projects from the list below, in order of preference. Applicants are invited to apply online.

For further information, please contact the PG tutor/selector in applied mathematics Dr Toby Wood.

#### Measuring gravity and accretion using pulsars

Measuring gravity and accretion using pulsarsNeutron stars are extremely dense cinders remaining after stellar explosions. They often have strong magnetic fields and rotate rapidly, and this combination often results in their appearing to pulsate with extreme regularity. We call these objects "pulsars", and their measurable rotation provides an opportunity to take precision measurements in some of the most extreme astrophysical environments accessible to observation.

This project will use existing observations and request and carry out new observations of pulsar systems. These observations will strongly constrain theoretical models of how matter falls onto neutron stars, and in fact probe the details of how gravity works - does it behave as Einstein predicted?

An understanding of the basics of astronomical observation and data analysis is required, as is an interest in understanding models of these phenomena and how to test them.

**Supervisor**: Dr Anne Archibald

#### The cosmic large-scale structure

The cosmic large-scale structureThe cosmic large-scale structure is the skeleton of matter on the largest scales in the Universe. Galaxies trace this large-scale skeleton of dark matter and form in large gravitationally bound dark matter structures. With major upcoming galaxy surveys like Euclid and LSST, we will be able to track the growth of structure through time across large volumes. This will provide a cosmic laboratory for testing cosmology, fundamental physics and astrophysics with the large-scale structure. To extract the maximum amount of information from galaxy surveys, we need a) accurate models for the gravitational dynamics of the dominant dark matter component, and b) powerful statistics that capture key aspects of gravitational clustering.

This PhD project will tackle these two intertwined challenges. First, we will use novel techniques to describe gravitational dark matter dynamics, for example using the quantum-classical correspondence. The goal is to develop new analytical and computational tools to solve for the time-evolution of dark matter and hunt for signatures of particular dark matter candidates. Second, we will develop clustering statistics that capture non-Gaussian properties of the late-time matter distribution. The idea is to use a sweet spot of simple statistics that are easy to measure, and can be accurately predicted into the nonlinear regime. With this, we will seek to improve the standard analysis relying on two-point statistics to obtain unique insights into cosmology, fundamental physics and astrophysics.

**Supervisor: Dr Cora Uhlemann**

#### Cosmological superfluids

Cosmological superfluidsOur understanding of dark matter remains elusive despite recent successes. A revolutionary recent idea suggests that dark matter could actually exist in the form of a gigantic cosmic superfluid, whose wave phenomena span thousands of light years. Such studies, still at their infancy, have so far only been conducted in a simple non-interacting condensate picture of scalar dark matter.

The aim of this ambitious, interdisciplinary project (supervised by both cosmologists and cold-atom theorists) is to utilise stochastic techniques established in ultracold atomic physics – where they offer precise quantitative agreement with controlled experiments – to understand such cosmic superfluids. We aim to go beyond simple models used to date and include, for example, effects of self-interactions, while also allowing the co-existence of dark matter particles in both a condensed and a gas-like, non-condensed component, separated by a phase transition.

The goal of this ambitious project is to understand Bose-Einstein condensation on cosmological scales, and notably in a setting where the phase transition is actually driven by self-collapse due to gravity. Our numerical simulations will map the relevant parameter space, and could eventually lead to placing constraints, in combination with astrophysical observations.

**Supervisors: Dr Gerasimos Rigopoulos** and **Professor Nick Proukakis**

#### Observational cosmology in a data-rich era

Observational cosmology in a data-rich eraCosmology is enjoying an era of unprecedented data abundance, with powerful observations already available and next-generation surveys on the immediate horizon. This wealth of data provides an exciting opportunity to pin down the nature of the mysterious dark energy which makes up 70% of the Universe. This PhD project will develop crucial, cutting-edge techniques for the analysis of modern cosmological survey data, and will apply these and other techniques to existing data in order to achieve new insight into the composition, history, and physical laws of our Universe.

**Supervisor: Dr Danielle Leonard**

#### The Big Bang and inflation

The Big Bang and inflationThe main goal of our cosmology programme is to search for direct observational signatures of early universe cosmological models, paying special attention to the role of quantum processes during a period of cosmological inflation. This PhD project covers topical aspects of the very early universe, including:

- the quantum theory of the big bang and the origin of time
- investigations of inflationary models with quantum gravity and Higgs bosons
- models which seek to explain the origin of primordial magnetic fields

**Supervisor:** Professor Ian Moss

#### The large scale structure of the universe

The large scale structure of the universeObservational cosmology has undergone a revolution in recent years due to the availability of high quality data from satellite observations. This makes it possible to attempt to find out what makes up the dominant form of matter in the universe. It also helps us understand the mysterious dark energy which is driving the universe apart. This PhD project attempts to fit models of dark matter and dark energy to observations of large scale structure using new approaches and new theories of dark energy.

**Supervisors: **Professor Ian Moss

#### Convective dynamo action at the solar surface

Convective dynamo action at the solar surfaceSolar magnetism plays a crucial role in determining the Sun’s influence on the environment of the Earth and the rest of the Solar System. Magnetic fields in the Sun, on both large and small scales, are generated and maintained by fluid motions within the solar interior. Whilst certain aspects of this dynamo process are well understood, there are many areas of uncertainty.

In this project, we will investigate the origin of the intermittent magnetic field distribution that is observed in regions of quiet Sun (areas of the solar surface that are devoid of large-scale magnetic features such as sunspots). We will test the hypothesis that these fields are generated locally by the near-surface convective motions.

Much of this project will be based upon high resolution numerical simulations of compressible magnetohydrodynamics, making use of an existing code. The output from this code will be used to address various important issues, including the extent to which the efficiency of the dynamo is influenced by the presence of convective motions on spatial scales that are larger than the small-scale 'granular' convective pattern that covers most of the solar surface.

The existence of one such convective scale, namely supergranulation, is well established. But the reality (or otherwise) of an additional 'mesogranular' scale is the subject of current controversy. By applying the observers' data analysis algorithms to our simulation data, we will attempt to resolve this issue over the course of the project.

Although no prior knowledge of solar physics will be assumed, a good understanding of fluid dynamics is essential.

**Supervisors: **Dr Paul Bushby and Dr Toby Wood

#### Convection in the Earth's core

Convection in the Earth's coreMotions of liquid iron in the Earth’s outer core are driven by thermal and compositional convection due to the cooling of the planet. An on-going debate in the geophysics community is whether the whole outer core is convecting or whether a region near the core-mantle boundary is stably stratified. In this project, we propose to study:

- the mechanisms by which a stratified layer could form at the top of Earth’s core
- the nature of the motions in this layer
- the layer thickness

The project is based on numerical simulations, which will be performed starting from existing numerical codes. A good understanding of fluid dynamics is essential. No prior knowledge of geophysics or computational modelling is required — the necessary training in these areas will be provided during the early stages of the project.

**Supervisors: **Dr Celine Guervilly and Dr Graeme Sarson

#### Galactic magnetic fields

Galactic magnetic fieldsWe describe Galaxies as islands in the Universe, each containing billions of stars.

Interstellar gas fills space between the stars. This gas is a complex hydrodynamical system involved in intense turbulent motions. It exhibits an exceedingly wide range of temperatures (from a few degrees on the Kelvin scale to several million degrees) and densities. It is permeated by magnetic fields and mixed with relativistic particles: cosmic rays.

The interstellar medium is especially rich in complexity in spiral galaxies, whose notable feature is rapid rotation. The interstellar medium feeds the formation of new stars. This largely controls the optical appearance of the host galaxy, and maintains magnetic fields. Despite the violently turbulent nature of the interstellar gas, these magnetic fields exhibit order at very large scales comparable to the size of the parent galaxy.

This project will focus on the origin of the large-scale magnetic fields in spiral galaxies. Recent developments in hydromagnetic dynamo theory have opened an opportunity to construct models that can be directly compared to astronomical observations, refined and perfected. Such a theoretical development is urgently required to plan and interpret observations with a new generation of powerful radio telescopes.

The work on the project will involve:

- development of the theory of galactic magnetic fields, based on numerical and analytical studies of the dynamo equations
- collection of relevant astronomical data, as required to adapt theoretical models to specific galaxies
- comparison of the theoretical predictions with radio astronomical observations

The work on the project will involve regular international contacts with both theoreticians and observational astronomers. It will involve a modest amount of numerical calculations (eg with Matlab or Fortran).

**Supervisors: **Professor Anvar Shukurov and Dr Andrew Fletcher

#### Buoyant magnetic fields in the Sun

Buoyant magnetic fields in the SunThe complex magnetic structures (coronal loops and prominences) that appear on the Sun's surface are believed to originate at the bottom of the solar convection zone. This is where the magnetic field is stretched and amplified until it becomes buoyant and rises to the surface. However, the details of this magnetic buoyancy process are poorly understood, and are strongly affected by the Sun's rotation and compressibility.

This project will combine analytical theory with high resolution numerical simulations, to determine how magnetic buoyancy operates under the conditions of the solar interior. We will also address the role that the rising magnetic field structures play in the Sun's overall dynamo process.

We will assume no prior knowledge of solar physics. A good understanding of fluid dynamics is essential.

**Supervisors: **Dr Toby Wood and Dr Paul Bushby

#### Neutron star magnetic fields

Neutron star magnetic fieldsNeutron stars are extremely dense and rapidly rotating objects. They have the strongest magnetic fields in the Universe. Regular stars are powered by nuclear reactions. Neutron stars are powered by their vast reservoirs of rotational and magnetic energy.

A neutron star has a solid outer crust surrounding a superfluid core. Within this core the rotation and magnetic field are "quantised" into thin filaments called vortices and fluxtubes.

This project will develop a model for the dynamics of the vortices in the star's core, and their interaction with the strong magnetic field. We will base our model on suitably modified fluid equations that take account of the superfluid nature of the core.

Basic knowledge of fluid dynamics is required, as well as interest in developing computational skills.

**Supervisors: **Dr Toby Wood and Professor Carlo F Barenghi

#### Stellar dynamics and evolution

Stellar dynamics and evolutionThe evolution of stars and their ultimate demise is affected by hydrodynamic processes occurring within their interiors throughout their lifetime.

Dynamical processes such as convection, rotation, waves and magnetism all greatly impact how these stars explode, chemically enrich the galactic environment and the properties of the stellar remnant.

This project will involve using multi-dimensional hydrodynamic processes to understand these dynamical processes and how they contribute to stellar evolution. Using this understanding, combined with observational constraints, we will develop one-dimensional prescriptions for use in standard stellar evolution models.

**Supervisor: ** Dr Tamara Rogers

#### Quantum droplets

Quantum dropletsLandmark experiments with atomic quantum gases since 2017 have demonstrated a new form of quantum matter - a quantum droplet. These droplets have several unusual properties:

- they are superfluid - meaning that they are free from viscosity and can support eternal flow
- they are self-supporting - like stars under their own gravity
- they also have such high particle density that quantum mechanical fluctuations and correlations, normally negligible in the gas phase, become significant

These unique features bring the droplets to the fore for studying exotic physics, such as laboratory analogs of neutron stars and highly-correlated quantum systems, and developing new technologies, such as ultra-precise sensors.

This timely project will develop computational and/or analytical models of the droplets to explore these state-of-the-art opportunities.

**Supervisors: ****Dr Nick Parker** and **Dr Tom Billam**

#### Quantum Mixtures

Quantum MixturesWe know well what happens when two classical systems interact: they can mix (e.g. milk and water), or phase-separate (e.g. oil and water). What happens then when two quantum fluids overlap? This depends crucially on their interaction strength, with the quantum nature of the many-body system setting new rules for their coupling – critically also depending on whether the atomic system is composed of bosonic, or fermionic, particles.

Motivated by experiments with a plethora of different mixtures of ultracold quantum gases, at temperatures below micro-Kelvin, the aim of this project is to study the static and dynamic properties of such multi-component systems.

Questions to be studied include:

- How do such quantum mixtures emerge from their classical systems across the phase transition?
- What difference does the bosonic, or fermionic, nature of the individual components play, and how does a double superfluid (i.e. a fluid in which both bosonic and fermionic components of a Bose-Fermi mixture are superfluid) differ from other mixtures?
- In particular, how does rotation influence the dynamics of quantum mixtures? (a question of indirect relevance to the cores of neutron stars)
- How does the presence of external (electromagnetic) coupling between different components influence the system’s properties?

Such questions will be addressed in close collaboration with European experimental groups, where such experiments are underway.

**Supervisors: Professor Nick Proukakis and Professor Carlo Barenghi **

#### Stochastic modelling of quantum matter

Stochastic modelling of quantum matterUnderstanding the behaviour of matter often requires the use of stochastic methods. We add random noise to numerical equations in a controlled way. This mimics the physical response of a system.

This arises across all aspects of modelling, from biological, chemical to physical systems. An obvious example is the random jitter of particles. Here, a random displacing noise is added to the otherwise stationary particle evolution. In the physical setting, the noise usually arises from the interaction of the object with a so-called “heat bath”. The object can exchange energy and particle number with the "heat bath".

Major advances in the last decades have led to a system of appropriate equations to model multi-particle quantum systems confined in appropriate geometries. These have been linked to many recent Nobel Prizes. Beyond a curiosity, such systems are accessible in the lab and promise to revolutionise our future quantum technologies.

The aims of this project are to:

- become familiar with the mathematical background, physical origin, and numerical implementation of such stochastic approaches
- use this knowledge to model cutting-edge experiments in at least two different physical systems which exhibit quantum effects on a macroscopic scale.

**Supervisors**: Professor Nick Proukakis

#### Is turbulence knotted?

Is turbulence knotted?Mathematicians, physicists and engineers have studied turbulence for more than a century. Almost all investigations into this fundamental problem of the natural sciences have concentrated the attention on two aspects:

- the geometry
- the dynamics of turbulence

Little attention has been paid, in comparison, to a third equally important aspect: the topology.

This is despite the fact that 19th century pioneers of fluid dynamics such as Kelvin and Helmholtz were already aware of the possibility of vortices becoming twisted, linked and knotted. Unfortunately, until recently, the only vortex structures which could be created in the laboratory were either very simple (such as vortex rings) or utterly complex (such as turbulence): the 'hydrogen atom' of topological complexity was missing. This situation suddenly changed in 2013, when Kleckner and Irvine, at the University of Chicago, showed that it is possible to create trefoil vortex knots under controlled and reproducible laboratory conditions. This breakthrough is now driving a great interest in the study of the topology of vortices and turbulence.

The project aims to:

- perform a numerical investigation of turbulent flows by solving the governing Euler or Navier-Stokes equations
- look for evidence of knotted structures.

The objectives are to:

- define and quantify the topological complexity of turbulence
- to explore the possibility of scaling laws

You should have an interests in fluid dynamics and methods of computational mathematics. You should be willing to learn tools from other relevant disciplines such as knot theory.

**Supervisors: **Professor Carlo F Barenghi and Dr Andrew Baggaley

#### Vortex lines as topological defects at phase transitions: micro big-bangs in superfluid helium

Vortex lines as topological defects at phase transitions: micro big-bangs in superfluid heliumThe progress of cosmology is limited as it cannot be studied in the laboratory. This remark motivates attempts to find physical systems which model some of the fundamental properties of the universe but are also experimentally accessible. One of such systems is the formation of topological defects at phase transitions. Below the temperature of approximately a milliKelvin, superfluid helium 3 exhibits a phase transition to a superfluid state characterised by the breaking of various symmetries that are good analogues to those broken after the Big Bang.

The experimental set up is the following. Superfluid helium 3 is locally heated by neutrons, creating a hot spot of normal liquid which expands and quickly cools back down to the superfluid state. Coherent superfluid regions of the liquid grow simultaneously, and, when they come in contact, the mismatch of the order parameter creates linear topological defects called vortex lines. It is thought that this process, called the Kibble-Zurek mechanism, in the context of cosmology, may be responsible for the formation of inhomogeneous large-scale structures, such as super clusters of galaxies. In liquid helium, the Kibble-Zurek mechanism has been observed experimentally at Helsinki and Grenoble.

To make better connection with theory, it is necessary to understand how the small region of vortex lines, a turbulent spot, diffuses in space and time: this is what we plan to do numerically. In particular, we want to find how strongly nonlinear processes such as vortex reconnections, which occur when two vortex lines collide with each other, affect the diffusion. Preliminary numerical experiment suggest that vortex rings evaporate away from the turbulent spot.

**Supervisor: **Professor Carlo F Barenghi

#### Vortex motion in atomic condensates

Vortex motion in atomic condensatesCurrent experiments with atomic condensates are concerned with the motion of vortices confined in a small geometry (typically discs or spheres). The small size of these systems means that this motion is affected by the boundaries.

The aim of this project is to gain more understanding of the behaviour of vortices and vortex clusters and possibly make connections with experiments. In particular, we shall use the point vortex method of classical Euler fluid dynamics and the nonlinear Gross-Pitaevksii equation to determine vortex trajectories and to study the chaotic properties of these vortex systems.

**Supervisor: **Professor Carlo F Barenghi

#### Atomtronic devices

Atomtronic devicesExperimental advances over the last decade are beginning to usher in an age of quantum devices, such as sensors and interferometers. They have the potential to surpass their classical predecessors in terms of:

- sensitivity
- reliability
- miniaturization

One paradigm for constructing such devices involves using ultracold atoms as the quantum element. This has led to the creation of "atomtronic" analogs of electronic (and optical) devices in which the electrons (or light) are replaced with a superfluid current of ultracold atoms. This atomic superfluid can be made to flow without viscosity through carefully shaped channels in a similar way to electricity flowing through circuits, and light travelling through photonic media such as optical fibres or nonlinear crystals. Owing to many-body quantum effects inherent in the atomic superfluid, atomtronics has possible applications in making ultra-sensitive, quantum-enhanced sensors and interferometers.

This project will develop computational and analytical models of novel atomtronic quantum interference devices. In particular, we will connect with recent developments in quantum electronics to generate and develop new proposals for atomtronic devices that exhibit quantum-enhanced performance.

S**upervisors:** Dr Tom Billam and Dr Clive Emary

#### Collective behaviour from cells to animals

Collective behaviour from cells to animalsLarge aggregates of living entities, from biological cells to animals, can exhibit rich and complex collective behaviour. This behaviour often arises from:

- simple actions of the individual members
- their interaction with their immediate neighbours and environment.

Striking examples of this in nature are bird flocks (most spectacularly the aerial display of huge numbers of starlings at dusk) and fish schools.

Collective behaviour also plays a key role on the microscopic scale of biological cells. In particular, in-vitro stem cells undergo complex dynamics as they evolve to form colonies and tissues. This process underlies future medical applications of stem cells for the controlled regeneration of biological tissue.

This project will develop a model for such emergent collective behaviour. You may look at the macroscopic domain of birds (Dr Baggaley / Dr Fletcher / Dr Gillespie) or the microscopic realm of stem cells (Dr Parker / Prof Shukurov). Comparison to experimental observations (for birds, this will be through live imaging taken as part of the PhD project; for stem cells, this will be obtained through a collaboration with state-of-the-art experiments at the Institute of Genetic Medicine) will help deduce the biological, physical and geometrical processes which govern these dynamics, and can be expected to shed new light on collective behaviour in these systems.

**Supervisors:** Dr Andrew Baggaley, Dr Andrew Fletcher, Dr Colin Gillespie, Dr Nick Parker and Professor Anvar Shukurov

#### Modelling of ecological systems

Modelling of ecological systemsWe live in an era of marked global change from climate change, deforestation and urbanisation. This has major implications for natural ecological systems such as plants and trees, insects, animals and coral reefs.

Drs Baggaley and Parker are working with government to understand the spread of tree disease from invasive species. Topical examples of this are the dieback fungus and borer beetle affected UK ash trees. Our work focusses on understanding the key factors governing the spread of the disease and how the damage might be mitigated.

Meanwhile Dr Emary is studying ecological networks – abstract representations of the interactions between species in an ecosystem. Key questions include the response of such networks to species loss and environmental change. This work is performed in in collaboration with field ecologists with applications in agriculture and climate-change mitigation.

Moreover, engineering ecological systems, such as bacterial colonies, may help us meet future challenges in energy provision and waste management. We are developing mathematical models for the bacterial colonies, so as to highlight conditions and strategies to optimising the efficiency of these processes.

**Supervisors:** Dr Andrew Baggaley, Dr Clive Emary and Dr Nick Parker

#### Improving medical imaging and diagnostics

Improving medical imaging and diagnosticsAdvances in medical imaging enable clinicians to probe the body with remarkable precision. Clinicians can gather extensive images and datasets. Such advances demand sophisticated mathematical techniques to extract clinically-relevant information. Our researchers are working with clinicians to contribute to these challenges. We work in a range of settings and use a range of analytical and computational methods.

One example is our work in monitoring the recovery of the cornea to stem cell therapies (Prof Shukurov). Following major trauma, the natural cellular structure of the cornea is destroyed. Stem cell therapies can lead to a recovery of this structure. Working with clinicians and microscope images of the eye, we are developing advanced methods to quantitatively characterise the cell structure. This means we can assess levels of damage and monitor the recovery process.

Another example is our work in studying medical ultrasound imaging within the body, in collaboration with clinical medical physics (Dr Parker). The refraction of the ultrasound (eg at tissue boundaries) leads to a geometrical distortion of the images which is not accounted for. This mathematical modelling may lead to the development of corrective strategies which may be translated into clinical devices.

**Supervisor:** Dr Andrew Baggaley, Dr Nick Parker and Professor Anvar Shukurov

#### Prehistoric population dynamics

Prehistoric population dynamicsPopulation dynamics is a well-established field of applied mathematics. It has a wide range of applications to biological and social systems. It has been especially successful in applications to prehistory where the fundamental features of the evolution of human populations were free from the unmanageable complications of politics, long-distance travel, etc. One of the most fascinating ages in the human prehistory was the Neolithic, the last period of the Stone Age. The defining feature of the Neolithic was the birth of agriculture and food production (as opposed to food-gathering and hunting). This resulted in:

- a more sedentary lifestyle
- the emergence of urbanism
- human societies as we now know them.

The Neolithic first appeared in the Near East and China (perhaps apart from other relatively minor sources) about 12-10 thousand years ago. It then spread across Europe and Asia. There are well-developed mathematical models of this process, but they suffer from several shortcomings:

- virtually all of them focus on a limited geographical region (eg Western Europe) and neglect any connections, spatial and temporal, with other regions
- it remains unclear how such environmental factors as topography, climate, soil quality, etc. affected the spread of the agriculturalists and their technologies.

This project aims to develop comprehensive mathematical models of the spread (and subsequent development) of the Neolithic in Eurasia. It will allow for the environmental effects, and take account of the multiple centres where agriculture was independently introduced.

Mathematical modelling, mostly based on numerical simulations, will be constrained by the archaeological and other evidence available, which we will interpret using statistical tools. The project will involve close contacts not only with other mathematicians but also with archaeologists. It may include participation in archaeological field trips and excavations, if desired.

**Supervisors:** Professor Anvar Shukurov and Dr Graeme Sarson

#### Resolved Mechanics In Continuum Models For Bio-Films And Flocs

Resolved Mechanics In Continuum Models For Bio-Films And FlocsExperimental techniques such as DNA sequencing and genome editing enable elegant studies into the inner workings of ‘simple’ single-cell organisms. Designer bugs that can eat plastics, digest toxic chemicals and produce bio-fuels are being extensively researched. In parallel with such lab based work there is also a growing body of mathematical and computational research directed at furthering our understanding of how micro-organisms organize their world, and how we might use this knowledge to better our own. This PhD project aligns with the mathematical/computational approach, and seeks to develop hi-fidelity models that accurately describe the collective behaviour and emergent properties of colonies of micro-organisms.

Many species of bacteria have evolved the ability to manufacture and secrete a sort of ‘glue’, commonly referred to as extra-polymeric substance (EPS). This substance serves a variety of purposes, not least it enables bacteria to ‘stick together’ and form colonies. These may adhere to surfaces as bio-films, or be suspended in fluids as bio-flocs. The EPS forms a protective matrix in which the cells can grow and divide. It also acts as a medium through which nutrients and cell metabolites can be transported, and by which cells may exchange chemical signals. In building mathematical models for the growth and behaviour of microbial colonies it is therefore important to take into account the role played by this EPS matrix.

A widely used modelling approach is that of agent-based descriptions; the individual cells in a colony are represented as discrete entities (agents) that grow, divide and interact with each other (and the EPS) through imposed biochemical and mechanical rules. Conceptually simple, and allowing detailed interactions to be prescribed relatively easily, this approach is designed for computer simulation. It has proved very effective for simulating the behaviour of colony formation and growth at small spatial scales, but as colony size increases the large number of cells is computationally prohibitive. (A 1mm square patch of biofilm will contain millions of individual cells). At larger scales, therefore, alternative modelling strategies are needed.

The main focus of this project will be on the development and application of population-based continuum models. Continuum models for bio-films, whilst not new, are perhaps less well developed and studied than agent-based models. In particular, the inclusion of microscale mechanical properties of EPS within continuum descriptions is an area where there is considerable modelling work to be done. Established modelling techniques developed in the context of multicomponent and granular continua will be adapted and applied to the type of biological media central to this project. Analysis of resulting models will involve both theoretical and numerical methods, and will require the development of some research codes.

As with all modelling, a key aspect of the work will be calibration and validation. This will draw on recent and on-going experimental and simulation studies: Experiments into the micro- and macro-scale viscoelastic properties of bio-films are in progress within the School of Engineering at Newcastle University. Data from these studies will inform parameter selection within proposed models. In addition, bio-film growth in channel flows is also being investigated experimentally, providing data that can be used to assess model predictions. The project will also have access to a recently developed, and mechanically detailed, agent-based simulation code, offering further reference data against which new continuum type models can be assessed.

The project would suit a mathematics or physics graduate with some background in continuum (fluid and/or solid) mechanics. Numerical work will require the development and use of computer codes, and programming experience/interest is necessary.

**Supervisor: Dr David Swailes**

#### Probabilistic Models For Particle Transport in Turbulent Flows

Probabilistic Models For Particle Transport in Turbulent FlowsFluid flows often transport material in the form of small solid particles, liquid droplets or gas bubbles. Sometimes all three at once; sand grains, oil drops and air bubbles in water for example. The particulate material may be present by design (spray atomization is an integral part of many engineering processes), or be unwelcome (micro-plastics in water systems, volcanic emissions etc.). In many of these multi-phase systems the underlying flow is turbulent, and the way in which the dispersed particulates interact with this flow is crucial to the overall transport process. Aerosols, for example, tend to cluster in high-strain/low-vorticity regions, which influences the rate at which these droplets coalesce.

One way to study particle dynamics in turbulent flows is through computer simulations: Based on an underlying particle equation of motion we can simulate the trajectories of many hundreds of thousands of individual particles, and thereby build up a statistical picture of the collective behaviour of the disperse phase, described in terms particle concentrations, average velocities, kinetic energies etc. These will inevitably depend on (and perhaps influence) the statistical properties of the turbulence.

A second approach is to develop models that govern directly how these statistical measures evolve in both space and time. A model that allows us to compute directly the statistical distribution of particles obviates the need to perform time-consuming particle tracking simulations (other than to test that the models are correct!). It is this second approach that forms the basis for the research in this project.

By treating particle equations of motion as stochastic ordinary differential equations (SDEs) we can formulate transport equations for probability density functions (pdfs) that describe the resulting, ensemble-based distributions of particle properties such as position and velocity. The SDEs are non-standard in that they incorporate stochastic processes and fields that are correlated both in time and in space. This feature reflects correlation structures inherent in turbulent flow, and has a profound effect on the form of the resulting pdf transport equations; previous mathematical analysis has identified a number of subtle challenges associated with both theoretical and numerical treatment of these pdf models. In this work we will consider how some of these issues may be addressed. Extended phase-space models (generalized Langevin equations) have been proposed. These eliminate non-Markovian features in the pdf models, but at the expense of higher-dimensionalities. Moreover, these extended models, as they now stand, are not capable of reproducing some key features of particle phase transport associated with preferential sampling and drift. We will aim to develop and assess strategies that address these important issues.

The project would suit a mathematics or physics graduate with a background in fluid dynamics modelling and/or stochastic analysis. Numerical work will require the development and use of computer codes, and programming experience/interest is highly desirable.

**Supervisor: Dr David Swailes**