The importance of coronary microcirculatory perfusion is highlighted by the severe impact of microvascular diseases such as diabetes and hypertension on heart function. Recently, highly-detailed three-dimensional (3D) data on ex vivo coronary microvascular structure have become available. However, hemodynamic information in individual myocardial capillaries cannot yet be obtained using current in vivo imaging techniques. In this thesis, a novel data-driven modelling framework is developed to predict tissue-scale flow properties from discrete anatomical data, which can in future be used to aid interpretation of coarse-scale perfusion imaging data in healthy and diseased states. Mathematical models are parametrised by the 3D anatomical data set of Lee (2009) from the rat myocardium, and tested using flow measurements in two-dimensional rat mesentery networks. Firstly, algorithmic and statistical tools are developed to separate branching arterioles and venules from mesh-like capillaries, and then to extract geometrical properties of the 3D capillary network. The multi-scale asymptotic homogenisation approach of Shipley and Chapman (2010) is adapted to derive a continuum model of coronary capillary fluid transport incorporating a non-Newtonian viscosity term. Tissue-scale flow is captured by Darcy's Law whose coefficient, the permeability tensor, transmits the volume-averaged capillary-scale flow variations to the tissue-scale equation. This anisotropic permeability tensor is explicitly calculated by solving the capillary-scale fluid mechanics problem on synthetic, stochastically-generated periodic networks parametrised by the geometrical data statistics, and a thorough sensitivity analysis is conducted. Permeability variations across the myocardium are computed by parametrising synthetic networks with transmurally-dependent data statistics, enabling the hypothesis that subendocardial permeability is much higher in diastole to compensate for severely-reduced systolic blood flow to be tested. The continuum Darcy flow model is parametrised by purely structural information to provide tissue-scale perfusion metrics, with the hypothesis that this model is less sensitive and more reliably parametrised than an alternative, estimated discrete network flow solution.

The main purpose of this work is to prove short-time structural stability of compressible vortex sheets with surface tension. The main result can be summarised as follows. Assume we start with an initial vortex-sheet configuration which consists of two inviscid fluids with density bounded below flowing smoothly past each other, where a strictly positive fixed coefficient of surface tension produces a surface tension force across the common interface, balanced by the pressure jump. We assume the fluids are modelled by the compressible Euler equations in three space dimensions with a very general equation of state relating the pressure, entropy and density in each fluid such that the sound speed is positive. Then, for a short time, which may depend on the initial configuration, there exists a unique solution of the equations with the same structure, that is, two fluids with density bounded below flowing smoothly past each other, where the surface tension force across the common interface balances the pressure jump. The mathematical approach consists of introducing a carefully chosen artificial viscosity-type regularisation which allows one to linearise the system so as to obtain a collection of transport equations for the entropy, pressure and curl together with a parabolic-type equation for the velocity. We prove a high order energy estimate for the non-linear equations that is independent of the artificial viscosity parameter which allows us to send it to zero. This approach loosely follows that introduced by Shkoller et al in the setting of a compressible liquid-vacuum interface. Although already considered by Shkoller et al, we also make some brief comments on the case of a compressible liquid-vacuum interface, which is obtained from the vortex sheets problem by replacing one of the fluids by vacuum, where it is possible to obtain a structural stability result even without surface tension.

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

When traditional linearised theory is used to study free-surface flows past a surface-piercing object or over an obstruction in a stream, the geometry of the object is usually lost, having been assumed small in one or several of its dimensions. In order to preserve the nonlinear nature of the geometry, asymptotic expansions in the low-Froude or low-Bond limits can be derived, but here, the solution invariably predicts a waveless free-surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In this thesis, we will apply exponential asymptotics to the study of two new problems involving nonlinear geometries. In the first, we examine the case of free-surface flow over a step including the effects of both gravity and surface tension. Here, we shall see that the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects, leads to the discovery of a remarkable new set of solutions. In the second problem, we study the waves produced by bluff-bodied ships in low-Froude flows. We will derive the analytical form of the exponentially small waves for a wide range of hull geometries, including single-cornered and multi-cornered ships, and then provide comparisons with numerical computations. A particularly significant result is our confirmation of the thirty-year old conjecture by Vanden-Broeck & Tuck (1977) regarding the impossibility of waveless single-cornered ships.

Tissue engineering has great potential as a method for replacing or repairing lost or damaged tissue. However, progress in the field to date has been limited, with only a few clinical successes despite active research covering a wide range of cell types and experimental approaches. Mathematical modelling can complement experiments and help improve understanding of the inherently complex tissue engineering systems, providing an alternative perspective in a more cost- and time-efficient manner. This thesis focusses on one particular experimental setup, a hollow fibre membrane bioreactor (HFMB). We develop a suite of mathematical models which consider the fluid flow, solute transport, and cell yield and distribution within a HFMB, each relevant to a different setup which could be implemented experimentally. In each case, the governing equations are obtained by taking the appropriate limit of a generalised multiphase model, based on porous flow mixture theory. These equations are then reduced as far as possible, through exploitation of the small aspect ratio of the bioreactor and by considering suitable parameter limits in the subsequent asymptotic analysis. The reduced systems are then either solved numerically or, if possible, analytically. In this way we not only aim to illustrate typical behaviours of each system in turn, but also highlight the dependence of results on key experimentally controllable parameter values in an analytically tractable and transparent manner. Due to the flexibility of the modelling approach, the models we present can readily be adapted to specific experimental conditions given appropriate data and, once validated, be used to inform and direct future experiments.

A validated model of water transport in the cerebral environment is both an ambitious and timely task; many brain diseases relate to imbalances in water regulation. From tumours to strokes, chronic or acute, transport of fluid in the brain plays a crucial role. The importance and complexity of the brain, together with the range of unmet clinical needs that are connected with this organ,make the current research a high-priority. One of the most paradoxical cerebral conditions, hydrocephalus, serves as an excellent metric for judging the success of anymodel developed. In particular, normal pressure hydrocephalus (NPH) is a paradoxical condition with no known cure and existing treatments display unacceptably high failure rates. NPH is considered to be a disease of old age, and like many such diseases, it is related to a change in the transport of fluid in the cerebral environment. This complex system ranges from organ-level transport to cellular membrane channels such as aquaporins; through integrating it in a novel mathematical framework, we suggest that the underlying logic of treatment methods may be misleading. By modelling the transport of cerebrospinal fluid (CSF) between the ventricular system, cerebral tissue and blood networks, we find that changes to the biophysical properties of the brain (rather than structural changes such as aqueduct obstruction) are capable of producing clinically relevant ventriculomegaly in the absence of any obstruction to CSF flowthrough the ventricular system. Specifically, the combination of increased leakiness and compliance of the capillary bed leads to the development of enlarged ventricles with a normal ventricular pressure, replicating clinical features of the presentation of NPH. These results, while needing experimental validation, imply that treatment methods like shunting, that are focussed on structural manipulation, may continue to fail at unacceptably high rates.

Cerebrospinal fluid (CSF) fills a system of cavities at the centre of the brain, known as ventricles, and the subarachnoid space surrounding the brain and the spinal cord. In addition, CSF is in free communication with the interstitial fluid of the brain tissue. Disturbances in CSF dynamics can lead to diseases that cause severe brain damage or even death. So-called infusion tests are frequently performed in the diagnosis of such diseases. In this type of test, changes in average CSF pressure are related to changes in CSF volume through infusion of known volumes of additional fluid. Traditionally, infusion tests are analysed with single compartment models, which treat all CSF as part of one compartment and balance fluid inflow, outflow and storage through a single ordinary differential equation. Poroelastic models of the brain, on the other hand, have been used to simulate spatial changes with disease, particularly of the ventricle size, on larger time scales of days, weeks or months. Wirth and Sobey (2008) developed a two-fluid poroelastic model of the brain in which CSF pressure pulsations are linked to arterial blood pressure pulsations. In this thesis, this model is developed further and simulation results are compared to clinical data. At first, the functional form of the compliance, which governs the storage of CSF in single compartment models, is examined by comparison of two different compliance models with clinical data. The derivations of a single-fluid and a two-fluid poroelastic model of the brain in spherical symmetry are laid out in detail and some of the parameters are related to the compliance functions considered earlier. The finite element implementation of the two-fluid model is described and finally simulation results of the average CSF pressure response and the pressure pulsations are compared to clinical data.

The photoactive component of a polymeric organic solar cell can be produced by drying a mixture consisting of a volatile solvent and non-volatile polymers. As the solvent evaporates, the polymers demix and self-assemble into microscale structures, the morphology of which plays a pivotal role in determining the efficiency of the resulting device. Thus, a detailed understanding of the physical mechanisms that drive and influence structure formation in evaporating solvent-polymer mixtures is of high scientific and industrial value. This thesis explores several problems that aim to produce novel insights into the dynamics of evaporating solvent-polymer mixtures. First, the role of compositional Marangoni instabilities in slowly evaporating binary mixtures is studied using the framework of linear stability theory. The analysis is non-trivial because evaporative mass loss naturally leads to a time-dependent base state. In the limit of slow evaporation compared to diffusion, a separation of time scales emerges in the linear stability problem, allowing asymptotic methods to be applied. In particular, an asymptotic solution to linear stability problems that have slowly evolving base states is derived. Using this solution, regions of parameter space where an oscillatory instability occurs are identified and used to formulate appropriate conditions for observing this phenomenon in future experiments. The second topic of this thesis is the use of multiphase fluid models to study the dynamics of evaporating solvent-polymer mixtures. A two-phase model is used to assess the role of compositional buoyancy and to examine the formation of a polymer-rich skin at the free surface. Then, a three-phase model is used to conduct a preliminary investigation of the link between evaporation and phase separation. Finally, this thesis explores the dynamics of a binary mixture that is confined between two horizontal walls using a diffusive phase-field model and its sharp-interface and thin-film approximations. We first determine the conditions under which a homogeneous mixture undergoes phase separation to form a metastable bilayer. We then present a novel mechanism for generating a repeating lateral sequence of alternating A-rich and B-rich domains from this bilayer.

In mathematical population genetics a multiple merger <i>n</i>-coalescent process, or <i>Λ</i> <i>n</i>-coalescent process, {<i>Πⁿ(t) t</i> ≥ 0} models the genealogical tree of a sample of size <i>n</i> (e.g. of DNA sequences) drawn from a large population of haploid individuals. We study various properties of <i>Λ</i> coalescents. Novel in our approach is that we introduce the partition lattice as well as cumulants into the study of functionals of coalescent processes. We illustrate the success of this approach on several examples. Cumulants allow us to reveal the relation between the tree height, <i>T_n</i>, respectively the total branch length, <i>L_n</i>, of the genealogical tree of Kingman’s <i>n</i>-coalescent, arguably the most celebrated coalescent process, and the Riemann zeta function. Drawing on results from lattice theory, we give a spectral decomposition for the generator of both the Kingman and the Bolthausen-Sznitman <i>n</i>-coalescent, the latter of which emerges as a genealogy in models of populations undergoing selection. Taking mutations into account, let <i>M_j</i> count the number of mutations that are shared by <i>j</i> individuals in the sample. The random vector (<i>M₁</i>,...,<i>M_n-1</i>), known as the site frequency spectrum, can be measured from genetical data and is therefore an important statistic from the point of view of applications. Fu worked out the expected value, the variance and the covariance of the marginals of the site frequency spectrum. Using the partition lattice we derive a formula for the cumulants of arbitrary order of the marginals of the site frequency spectrum. Following another line of research, we provide a law of large numbers for a family of <i>Λ</i> coalescents. To be more specific, we show that the process {<i>#Πⁿ(t), t</i> ≥ 0} recording the number <i>#Πⁿ(t)</i> of individuals in the coalescent at time <i>t</i>, coverges, after a suitable rescaling, towards a deterministic limit as the sample size <i>n</i> grows without bound. In the statistical physics literature this limit is known as a hydrodynamic limit. Up to date the hydrodynamic limit was known for Kingman’s coalescent, but not for other <i>Λ</i> coalescents. We work out the hydrodynamic limit for beta coalescents that come down from infinity, which is an important subclass of the <i>Λ</i> coalescents.

In vitro tissue engineering is a method for developing living and functional tissues external to the body, often within a device called a bioreactor to control the chemical and mechanical environment. However, the quality of bone tissue engineered products is currently inadequate for clinical use as the implant cannot bear weight. In an effort to improve the quality of the construct, hydrostatic pressure, the pressure in a fluid at equilibrium that is required to balance the force exerted by the weight of the fluid above, has been investigated as a mechanical stimulus for promoting extracellular matrix deposition and mineralisation within bone tissue. Thus far, little research has been performed into understanding the response of bone tissue cells to mechanical stimulation. In this thesis we investigate an in vitro bone tissue engineering experimental setup, whereby human mesenchymal stem cells are seeded within a collagen gel and cultured in a hydrostatic pressure bioreactor. In collaboration with experimentalists a suite of mathematical models of increasing complexity is developed and appropriate numerical methods are used to simulate these models. Each of the models investigates different aspects of the experimental setup, from focusing on global quantities of interest through to investigating their detailed local spatial distribution. The aim of this work is to increase understanding of the underlying physical processes which drive the growth and development of the construct, and identify which factors contribute to the highly heterogeneous spatial distribution of the mineralised extracellular matrix seen experimentally. The first model considered is a purely temporal model, where the evolution of cells, solid substrate, which accounts for the initial collagen scaffold and deposited extracellular matrix along with attendant mineralisation, and fluid in response to the applied pressure are examined. We demonstrate that including the history of the mechanical loading of cells is important in determining the quantity of deposited substrate. The second and third models extend this non-spatial model, and examine biochemically and biomechanically-induced spatial patterning separately. The first of these spatial models demonstrates that nutrient diffusion along with nutrient-dependent mass transfer terms qualitatively reproduces the heterogeneous spatial effects seen experimentally. The second multiphase model is used to investigate whether the magnitude of the shear stresses generated by fluid flow, can qualitatively explain the heterogeneous mineralisation seen in the experiments. Numerical simulations reveal that the spatial distribution of the fluid shear stress magnitude is highly heterogeneous, which could be related to the spatial heterogeneity in the mineralisation seen experimentally.