## Continuum Mechanics

The Continuum Mechanics group brings together researchers in fluid and solid mechanics - focussing on both the underlying mathematical theories that accurately describe a wide range of materials, as well as the application of these theories to real-world systems. Our research spans a vast array of lengthscales, from models of dense swarms of swimming bacteria, through to the development of an accurate model of both the fluid and elastic behaviour of the heart, to the flow of vast glaciers or the fluid dynamics of the sun.

Many members of this group are also involved in SofTMech - a large and established Centre of Mathematics for Healthcare, which has had support from the EPSRC since 2016. The Centre consists of multiple academic, clinical and industrial partners and combines extensive expertise in mathematical, statistical and computational modelling of cardiac and cancer physiology and disease.

# Information about

## Staff

#### Dr Joseph Cousins : Research Assistant

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**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

#### Prof Nicholas A Hill : Simson Chair

Biological and physiological fluid dynamics; bioconvection; physiological pulse propagation;; Soft tissue mechanics; aneurysms; aortic dissection; gall bladder pain; upscaling; Random walk models for movement of micro-organisms and animals; spatial point processes in plant ecology

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research staff:** Jay MacKenzie, Scott Richardson

**Research students:** Andrew Brown, Sathish Kumar, Roxanna Barry, Laura Miller

**Postgraduate opportunities:** A coupled cardiovascular-respiration model for mechanical ventilation

#### Dr Katarzyna Kowal : Lecturer

Mathematical modelling and analysis of a broad range of industrial, biological, environmental and geophysical processes involving fluid flow and/or phase change.; Mathematical modelling and analysis of a broad range of industrial, biological, environmental and geophysical processes involving fluid flow and/or phase change.

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

#### Dr Wenguang Li : Postdoctoral Research Fellow

Biomechanics; modelling of gallbladder; non-linear finite strain modelling

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems

**Supervisor:** Xiaoyu Luo

#### Prof Xiaoyu Luo : Professor of Applied Mathematics

Biomechanics; fluid-structure interactions; mathematical biology ; solid mechanics

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research staff:** Liuyang Feng, Hao Gao, Debao Guan, Wenguang Li, Qingying Shu, Xin Zhuan, Jay MacKenzie

**Research students:** Ahmed Mostafa Abdelhady Ismaeel, Yingjie Wang

#### Dr David MacTaggart : Lecturer

Theoretical fluid dynamics, Magnetohydrodynamics, Magnetic topology

**Member of other research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research students:** Jamie Quinn, Parag Gupta

**Postgraduate opportunities:** Magnetic helicity as the key to dynamo bistability , Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD), Stellar atmospheres and their magnetic helicity fluxes

#### Prof Nigel Mottram : Professor of Applied Mathematics

My research interests are in the mathematical modelling of real-world systems, generally focussing on those that include the dynamics of non-Newtonian fluids. I am particularly interested in anisotropic fluids such as liquid crystals, where viscoelasticity is an important consideration, as is their behaviour under electric fields.

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research student:** Parna Mandal

#### Dr Raymond W Ogden : George Sinclair Chair

Nonlinear elasticity; biomechanics

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems

**Research staff:** Andrey Melnik

#### Dr Raimondo Penta : Lecturer

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research students:** Tahani Al Sariri, Laura Miller, Andrew Brown

#### Dr Ariel Ramirez Torres : Lecturer

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

#### Dr Scott Richardson : Research Associate

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**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisors:** Andrew Baggaley, Nicholas A Hill

#### Dr Steven Roper : Lecturer

I am interested in applications of continuum mechanics to phenomena in materials science. In particular problems involving solid mechanics and surface science (the combined chemical and mechanical equilibrium of droplets of fluid in contact with soluble substrates); crystal growth; pattern formation (block co-polymers).; I am interested in applications of fluid mechanics to industrial and geophysical problems. In particular in fluid-driven fracture (the interaction of flow fluid with fracture mechanics); porous media (particularly the reactive porous media found in the solidification of pure and alloyed materials - mushy layers); compositional convection; low Reynolds number flow (thin films and free surface flows); the behaviour of foams.

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research students:** Bushra Al-Ghabshi, Andrew Brown

#### Prof Radostin Simitev : Professor of Applied Mathematics

Reaction-diffusion equations; Excitable systems; Mathematical models of cardiac electrical excitation; Thermal convection in rotating systems; MHD and dynamo theory

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research staff:** Peter Mortensen

**Research students:** Muhamad Bin Noor Aziz, Parag Gupta, Antesar Mohammed Al Dawoud, Tahani Al Sariri, Jamie Quinn

**Postgraduate opportunities:** Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD), Fast-slow asymptotic analysis of cardiac excitation models, Numerical simulations of planetary and stellar dynamos, Efficient asymptotic-numerical methods for cardiac electrophysiology, Stellar atmospheres and their magnetic helicity fluxes, Personal Website

#### Dr Peter Stewart : Senior lecturer

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**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Research students:** Roxanna Barry, Ahmed Mostafa Abdelhady Ismaeel, Gordon McNicol, Ifeanyi Onah Sunday

**Postgraduate opportunities:** Mathematical models of vasculogenesis, A coupled cardiovascular-respiration model for mechanical ventilation , Radial foam fracture, Continuous production of solid metal foams

#### Dr Robert Teed : Lecturer

**Member of other research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Postgraduate opportunities:** Force balances in planetary cores and atmospheres, Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD), Stellar atmospheres and their magnetic helicity fluxes

#### Dr Stephen J Watson : Lecturer

; The application of new mathematical ideas and new computational paradigms to material science, with an emphasis on self-assembling nano-materials; analysis and numerical analysis of partial differential equations arising in Continuum Physics; Material Science and Geometry.

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems, Analysis

#### Dr Wei Zhang : Lecturer

Bayesian data analysis, Ecological statistics, Statistical computing ;

**Member of other research groups:** Statistics and Data Analytics, Continuum Mechanics - Modelling and Analysis of Material Systems

#### Dr Xin Zhuan : Research Associate

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**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems

**Supervisor:** Xiaoyu Luo

## Postgraduates

#### Bushra Al-Ghabshi : PhD Student

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems

**Supervisor:** Steven Roper

#### Andrew Brown : PhD Student

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**Research Topic:** Multiscale Modelling of Tissue Tearing

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisors:** Nicholas A Hill, Raimondo Penta, Steven Roper

#### Parag Gupta : PhD Student

**Research Topic:** Modelling of stellar differential rotation and related dynamo
properties

**Member of other research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisors:** Radostin Simitev, David MacTaggart

#### Sathish Kumar : PhD Student

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**Research Topic:** Optimisation of stent devices to treat dissected aorta

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisor:** Nicholas A Hill

#### Parna Mandal : PhD Student

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**Research Topic:** Optimising antibiotic release from medical implants to counteract
biofilm formation

**Member of other research groups:** Continuum Mechanics - Modelling and Analysis of Material Systems, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisor:** Nigel Mottram

#### Gordon McNicol : PhD Student

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**Research Topic:** A mathematical model for nanokicking

**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisor:** Peter Stewart

#### Laura Miller : PhD Student

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**Member of other research groups:** Mathematical Biology, Continuum Mechanics - Modelling and Analysis of Material Systems

**Supervisors:** Raimondo Penta, Nicholas A Hill

#### Jamie Quinn : PhD Student

**Member of other research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

**Supervisors:** David MacTaggart, Radostin Simitev

## Postgraduate opportunities

### Continuous production of solid metal foams (PhD)

**Supervisors:** Peter Stewart

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems

Porous metallic solids, or solid metal foams, are exceedingly useful in many engineering applications, as they can be manufactured to be strong yet exceedingly lightweight. However, industrial processing methods for producing such foams are problematic and unreliable, and it is not currently possible to control the porosity distribution of the final product a priori.

This project will consider a new method of solid foam production, where bubbles of gas are introduced continuously into a molten metal flowing through a heat exchanger; foaming and solidification then occur almost simulatanously, allowing the foam structure to be controlled pointwise. The aim of this project is to construct a simple mathematical model for a gas bubble moving in a liquid filled channel ahead of a solidification front, to predict optimal conditions whereby the gas bubble is drawn toward the phase boundary, hence forming a porous solid.

This project will require some background in fluid mechanics and a combination of analytical and numerical techniques for solving partial differential equations.

### Radial foam fracture (PhD)

**Supervisors:** Peter Stewart

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems

Gas-liquid foams are a useful analgoue of crystalline atomic solids. 2D foam fracture has been used to study the mechanisms of fracture in metals. A two-dimenisonal network model (formed from a large system of differential equations) has recently been produced to study foam fracture in a rectangular channel which is pressurised along one edge. This model has helped to explain the origin of the velocity gap (a range of inadmissable steady fracture velocities), observed both in foam fracture experiments and in atomistic simulations of brittle fracture. This project will apply this network modelling approach to study radial foam fracture in a Hele-Shaw cell, to mimick recent experiments. This system has strong similarity to radial Saffmann-Taylor fingering, where fingering has been observed when a less viscous fluid displaces a more viscous fluid in a confined geometry. This project will involve studying systems of ordinary and partial differential equations using both numerical and analytical methods.

### Numerical simulations of planetary and stellar dynamos (PhD)

**Supervisors:** Radostin Simitev

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Using Fluid Dynamics and Magnetohydrodynamics to model the magnetic fields of the Earth, planets, the Sun and stars. Involves high-performance computing.

### Mathematical models of vasculogenesis (PhD)

**Supervisors:** Peter Stewart

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Continuum Mechanics - Modelling and Analysis of Material Systems, Mathematical Biology

Vasculogenesis is the process of forming new blood vessels from endothelial cells, which occurs during embryonic development. Viable blood vessels facilitate tissue perfusion, allowing the tissue to grow beyond the diffusion-limited size. However, in the absence of vasculogenesis, efforts to engineer functional tissues (eg for implantation) are restricted to this diffusion-limited size. This project will investigate mathematical models for vasculogenesis and explore mechanisms to stimulate blood vessel formation for *in vitro *tissues. The project will involve collaboration with Department of Biological Engineering at MIT, as part of the SofTMech^{MP} project.

### A coupled cardiovascular-respiration model for mechanical ventilation (PhD)

**Supervisors:** Peter Stewart, Nicholas A Hill

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics, Mathematical Biology

Mechanical ventilation is a clinical treatment used to draw air into the lungs to facilitate breathing, used in treatment of premature babies with respiratory distress syndrome and in the treatment of severe Covid pneumonia. The aim is to oxygenate the blood while simultaneously removing unwanted by-products. However, over-inflation of the lungs can reduce the blood supply to the gas exchange surfaces, leading to a ventilation-perfusion mis-match. This PhD project will give you the opportunity to develop a mathematical model to describe the coupling between blood flow in the pulmonary circulation and air flow in the lungs (during both inspiration and expiration). You will devise a coupled computational framework, capable of testing patient-specific ventilation protocols. This is an ideal project for a postgraduate student with an interest in applying mathematical modelling and image analysis to predictive healthcare. The project will give you the opportunity to join a cross-disciplinary Research Hub that aims to push the boundaries of quantitative medicine and improve clinical decision making using innovative mathematical and statistical modelling.

### Observationally-constrained 3D convective spherical models of the solar dynamo (Solar MHD) (PhD)

**Supervisors:** Radostin Simitev, David MacTaggart, Robert Teed

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Solar magnetic fields are produced by a dynamo process in the Solar convection zone by turbulent motions acting against Ohmic dissipation. Solar magnetic activity affects nearEarth space environment and can harm modern technology and endanger human health. Further, Solar magnetism poses fundamental physical and mathematical problems, e.g. about the nature of plasma turbulence and the topology of magnetic field generation. Current models of the global Solar dynamo fall in two classes (a) mean-field dynamos (b) convection-driven dynamos. The mean-field models are only phenomenological as they replace turbulent interactions by ad-hoc source and quenching terms. On the other hand, spherical convection-driven dynamo models are derived from basic principles with minimal assumptions and potentially offer true predictive power; these can also be extended to other stars and giant planets. However, at present, convection driven dynamo models operate in a wrong dynamical regime and have limited success in reproducing a number of important 1 observations including (a) the sunspot cycle period, polarity reversals and the sunspot butterfly diagram, (b) the poleward migration of diffuse surface magnetic fields, (c) the polar field strength and phase relationships between poloidal/toroidal components. The aims of this project are to (a) develop a three-dimensional convection-driven Solar dynamo model constrained by assimilation of helioseismic data, and (b) start to use the model to estimate turbulent properties that determine the internal dynamics and activity cycles of the Sun.

### Force balances in planetary cores and atmospheres (PhD)

**Supervisors:** Robert Teed, Radostin Simitev

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Current dynamo simulations are run, not under the conditions of planetary cores and atmospheres, but in a regime idealised for computations. To forecast changes in planetary magnetic fields such as reversals and dynamo collapse, it is vital to understand the actual fluid dynamics of these regions.

The aim of this project is to produce simulations of planetary cores and atmospheres with realistic force balances and, in doing so, understand how such force balances arise and affect the dynamics of the flow. Force balances control many aspects of the fluid dynamics, and hence the dynamo process itself, including the size of flow structure, the buoyancy flux and zonal flows, so an understanding of the force balance available in various planetary cores and atmospheres is vital for understanding their dynamo processes. To achieve this the project will use a different technique to that typically used in dynamo simulations. The approach is to perform global simulations in a spherical shell with a magnetic field imposed by explicitly setting a component (or components) of the field at one of the boundaries. Within the interior the field is evaluated as normal using the induction equation. This set-up amounts to a model of magnetoconvection where the dynamics of the flow and magnetic field can be studied independently of the dynamo mechanism.

### Magnetic helicity as the key to dynamo bistability (PhD)

**Supervisors:** David MacTaggart, Radostin Simitev

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

The planets in the solar system exhibit very different types of large-scale magnetic field.The Earth has a strongly dipolar field, whereas the fields of other solar system planets, such as Uranus and Neptune, are far more irregular. Although the different physical compositions of the planets of the solar system will influence the behaviour of the large-scale magnetic fields that they generate, the morphology of planetary magnetic fields can depend on properties of dynamos common to all planets. Here, we refer to an important and recent discovery from dynamo simulations. Remarkably, two very different types of chaotic dipolar dynamo solutions have been found to exist over identical values of the basic parameters of a generic model of convection-driven dynamos in rotating spherical shells. The two solutions mentioned above can be characterised as ‘mean dynamos’, MD, where a strong poloidal field dominates and ‘fluctuating dynamos’, FD, where the poloidal component is weaker and the large-scale field can be described as multipolar. Although these two states have been shown to be bistable (co-exist) for a wide range of identical parameters, it is not clear how a particular state, MD or FD, is chosen and how/when one state can change to the other. Some of the bifurcations of such states has been investigated, but a deep understanding of the dynamics that cause the bifurcations remains to be developed. Since the magnetic topology of MD and FD states are fundamentally different, an important part of this project will be to probe the nature of MD and FD states by studying magnetic helicity, a magnetohydrodynamic invariant that combines information on the topology of the magnetic field with the magnetic flux. The role of magnetic helicity and other helicities (e.g. cross helicity) is currently not well understood in relation to MD and FD states, but these quantities are conjectured to be very important in the development of MD and FD states.

Bistability is also related to a very important phenomenon in dynamos - global field reversal. A strongly dipolar (MD) field can change to a transitional multipolar (FD) state before a reversal and then settle into another dipolar equilibrium (of opposite polarity) again after the reversal.This project aims to develop a coherent picture of how bistability operates in spherical dynamos. Since bistability is a fundamental property of dynamos, a characterisation of how bistable solutions form and develop is key for any deep understanding of planetary dynamos and, in particular, could be crucial for understanding magnetic field reversals.

### Stellar atmospheres and their magnetic helicity fluxes (PhD)

**Supervisors:** Simon Candelaresi, Radostin Simitev, David MacTaggart, Robert Teed

**Relevant research groups:** Continuum Mechanics - Fluid Dynamics and Magnetohydrodynamics

Our Sun and many other stars have a strong large-scale magnetic field with a characteristic time variation. We know that this field is being generated via a dynamo mechanism driven by the turbulent convective motions inside the stars. The magnetic helicity, a quantifier of the field’s topology, is and essential ingredient in this process. In turbulent environments it is responsible for the inverse cascade that leads to the large-scale field, while the build up of its small-scale component can quench the dynamo.

In this project, the student will study the effects of magnetic helicity fluxes that happen below the stellar surface (photosphere), within the stellar atmosphere (chromosphere and corona) and between these two layers. This will be done using two-dimensional mean field simulations that allow parameter studies for different physical parameters. A fully three-dimensional model of a convective stellar wedge will then be used to provide a more detailed picture of the helicity fluxes and their effect on the dynamo. Using recent advancements that allow us to extract surface helicity fluxes from solar observations, the student will make use of observations to verify the simulation results. Other recent observational results on the stellar magnetic helicity will be used to benchmark the findings.

## Multi-scale modelling of living tissues

Biological material growth and remodelling is of great importance in biomechanics.