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A weakly nonlinear stability analysis was conducted for the flow induced in an incompressible, Newtonian, viscous fluid lying between two infinite parallel plates which form a channel. The plates are oscillating synchronously in simple harmonic motion. The disturbed velocity of the flow was written in the form of a series in powers of a parameter which is a measure of the distance away from the linear theory neutral conditions. The individual terms of this series were decomposed using Floquet theory and Fourier series in time. The equations at second order and third order in were derived, and solutions for the Fourier coefficients were found using pseudospectral methods for the spatial variables. Various alternative methods of computation were applied to check the validity of the results obtained. The Landau equation for the amplitude of the disturbance was obtained, and the existence of equilibrium amplitude solutions inferred. The values of the coefficients in the Landau equation were calculated for the nondimensional channel half-widths h for the cases h = 5, 8, 10, 12, 14 and 16. It was found that equilibrium amplitude solutions exist for points in wavenumber Reynolds number space above the smooth portion of the previously determined linear stability neutral curve in all the cases examined. Similarly, Landau coefficients were calculated on a special feature of the neutral curve (called a ???finger???) for the case h = 12. Equilibrium amplitude solutions were found to exist at points inside the finger, and in a particular region outside near the top of the finger. Traces of the x-components of the disturbance velocities have been presented for a range of positions across the channel, together with the size of the equilibrium amplitude at these positions. As well, traces of the x-component of the velocity of the disturbed flow and traces of the velocity of the basic flow have been given for comparison at a particular position in the channel.

Motivated by applications in tissue engineering, this thesis is concerned with the flow through and around a free-moving porous tissue construct (TC) within a high-aspect-ratio vessel (HARV) bioreactor. We formalise and extend various results for flow within a Hele-Shaw cell containing a porous obstacle. We also consider the impact of the flow on related nutrient transport problems. The HARV bioreactor is a cylinder with circular cross-section which rotates about its axis at a constant rate, and is filled with a nutrient-rich culture medium. The porous TC is modelled as a rigid porous cylinder with circular cross-section and is fully saturated with the fluid. We formulate the flow problem for a porous TC (governed by Darcy's equations) within a HARV bioreactor (governed by the Navier-Stokes equations). We couple the two regions via appropriate interfacial conditions which are derived by consideration of the intricate boundary-layer structure close to the TC surface. By exploiting various small parameters, we simplify the system of equations by performing an asymptotic analysis, and investigate the resulting system for the flow due to a prescribed TC motion. The motion of the TC is determined by analysis of the force and torque acting upon it, and the resulting equations of motion (which are coupled to the flow) are investigated. The short-time TC behaviour is periodic, but we are able to study the long-time drift from this periodic solution by considering the effect of inertia using a multiple-scale analysis. We find that, contrary to received wisdom, inertia affects TC drift on a similar timescale to tissue growth. Finally, we consider the advection of nutrient through the bioreactor and TC, and investigate the problem of nutrient advection-diffusion for a simplified model involving nutrient uptake.

The behaviour of free-surface gravity waves on small Froude number fluid flow past some obstacle cannot be determined using ordinary asymptotic power series methods, as the amplitude of the waves is exponentially small. An exponential asymptotic method is used by Chapman and Vanden-Broeck (2006) to consider the problem of two-dimensional, steady flow past a submerged obstacle in the small Froude number limit, finding that a steady downstream wavetrainis switched on rapidly across a curve known as a Stokes line. Here, equivalent wavetrains on three-dimensional and unsteady flow configurations are considered, and Stokes switching causedby the interaction between exponentially small free-surface components is shown to play an important role in both cases. The behaviour of free-surface gravity waves is introduced by considering the problem of steady free-surface flow due to a line source. A steady wavetrain is shown to exist in the far field, and the behaviour of these waves is compared to existing numerical results. The problem of unsteady flow over a step is subsequently investigated, with the flow behaviour formulated in terms of Lagrangian coordinates so that the position of the free surface is fixed. Initially, the problem is linearized in the step-height, and the steady wavetrain is shown to spread downstream over time. The position of the wavefront is determined by considering the full Stokes structure present in the problem. The equivalent fully-nonlinear problem is then considered, with the position of the Stokes lines, and hence the wavefront, being determined numerically. Finally, linearized three-dimensional free-surface flow past an obstacle is considered in both the steady and unsteady case. The surface is shown to contain downstream longitudinal and transverse waves. These waves are shown to propagate downstream in the unsteady case, with the position of the wavefront again determined by considering the full Stokes structure of the problem.

The ability of cells to move through their environment and spread on surfaces is fundamental to a host of biological processes; including wound healing, growth and immune surveillance. Controlling cell motion has wide-ranging potential for medical applications; including prevention of cancer metastasis and improved colonisation of clinical implants. The relevance of the topic coupled with the naturally arising interplay of biomechanical and biochemical mechanisms that control cell motility make it an exciting problem for mathematical modellers. Two-phase flow models have been widely used in the literature to model cell motility; however, little is known about the mathematical properties of this framework. The majority of this thesis is dedicated to improving our understanding of the two-phase flow framework. We first present the simplest biologically plausible two-phase model for a cell crawling on a flat surface. Stability analyses and a numerical study reveal a number of features relevant to modelling cell motility. That these features are present in such a stripped-down two-phase flow model is notable. We then proceed to investigate how these features are altered in a series of generalisations to the minimal model. We consider the effect of membrane-regulated polymerization of the cell's actin network, the effect of describing the network as viscoelastic, and the effect of explicitly modelling myosin, which drives contraction of the actin network. Validation of hydrodynamical models for cell crawling and spreading requires data on cell shape. The latter part of the thesis develops an image processing routine for extracting the three-dimensional shape of cells settling on a flat surface from confocal microscopy data. Models for cell and droplet settling available in the literature are reviewed and we demonstrate how these could be compared to our cell data. Finally, we summarise the key results and highlight directions for future work.

Three different models, relating to the study of microswimmers immersed in a low Reynolds number fluid, are presented. The underlying, mathematical concepts employed in each are developed using singularity methods of Stokes flow. The first topic concerns the motility of an artificial, three-sphere microswimmer with prescribed, non-reciprocal, internal forces. The swimmer progresses through a low Reynolds number, nonlinear, viscoelastic medium. The model developed illustrates that the presence of the viscoelastic rheology, when compared to a Newtonian environment, increases both the net displacement and swimming efficiency of the microswimmer. The second area concerns biological microswimming, modelling a sperm cell with a hyperactive waveform (vigorous, asymmetric beating), bound to the epithelial walls of the female, reproductive tract. Using resistive-force theory, the model concludes that, for certain regions in parameter space, hyperactivated sperm cells can induce mechanical forces that pull the cell away from the wall binding. This appears to occur via the regulation of the beat amplitude, wavenumber and beat asymmetry. The next topic presents a novel generalisation of slender-body theory that is capable of calculating the approximate flow field around a long, thin, slender body with circular cross sections that vary arbitrarily in radius along a curvilinear centre-line. New, permissible, slender-body shapes include a tapered flagellum and those with ribbed, wave-like structures. Finally, the detailed analytics of the generalised, slender-body theory are exploited to develop a numerical implementation capable of simulating a wider range of slender-body geometries compared to previous studies in the field.

Although the leading-order motion of waves is periodic - in other words backwards and forwards - many types of waves including those driven by gravity induce a mean flow as a higher-order effect. It is the induced mean flow of three types of gravity waves that this thesis examines: surface (part I), internal (part II) and interfacial gravity waves (part III). In particular, this thesis examines wave groups. Because they transport energy, momentum and other tracers, wave-induced mean flows have important consequences for climate, environment, air traffic, fisheries, offshore oil and other industries. In this thesis perturbation methods are used to develop a simplified understanding of the physics of the induced mean flow for each of these three types of gravity wave groups. Leading-order estimates of different transport quantities are developed. For surface gravity wave groups (part I), the induced mean flow consists of two compo- nents: the Stokes drift dominant near the surface and the Eulerian return flow acting in the opposite direction and dominant at depth. By considering subsequent orders in a separation of scales expansion and by comparing to the Fourier-space solutions of Longuet-Higgins and Stewart (1962), this thesis shows that the effects of frequency dis- persion can be ignored for deep-water waves with realistic bandwidths. An approximate depth scale is developed and validated above which the Stokes drift is dominant and below which the return flow wins: the transition depth. Results are extended to include the effects of finite depth and directional spreading. Internal gravity wave groups (part II) do not display Stokes drift, but a quantity analogous to Stokes transport for surface gravity waves can still be developed, termed the “divergent- flux induced flow” herein. The divergent-flux induced flow it itself a divergent flow and induces a response. In a three-dimensional geometry, the divergent-flux induced flow and the return flow form a balanced circulation in the horizontal plane with the former transporting fluid through the centre of the group and the latter acting in the opposite direction around the group. In a two-dimensional geometry, stratification inhibits a balanced circulation and a second type of waves are generated that travel far ahead and in the lee of the wave group. The results in the seminal work of Bretherton (1969b) are thus validated, explicit expressions for the response and return flow are developed and compared to numerical simulations in the two-dimensional case. Finally, for interfacial wave groups (part III) the induced mean flow is shown to behave analogously to the surface wave problem of part I. Exploring both pure interfacial waves in a channel with a closed lid and interacting surface and interfacial waves, expressions for the Stokes drift and return flow are found for different configurations with the mean set-up or set-down of the interface playing an important role.

Turbidity currents are one of the primary means of transport of sediment in the ocean. They are fast-moving, destructive fluid flows which are able to entrain sediment from the seabed and accelerate downslope in a process known as `ignition'. In this thesis, we investigate one particular model for turbidity currents; the `Parker model' of Parker, Pantin and Fukushima (1986), which models the current as a continuous sediment stream and consists of four equations for the depth, velocity, sediment concentration and turbulent kinetic energy of the flow. We propose two reduced forms of the model; a one-equation velocity model and a two-equation shallow-water model. Both these models give an insight into the dynamics of a turbidity current propagating downstream and we find the slope profile to be particularly influential. Regions of supercritical and subcritical flow are identified and the model is solved through a combination of asymptotic approximations and numerical solutions. We next consider the dynamics of the four-equation model, which provides a particular focus on Parker's turbulent kinetic energy equation. This equation is found to fail catastrophically and predict complex-valued solutions when the sediment-induced stratification of the current becomes large. We propose a new `transition' model for turbulent kinetic energy which features a switch from an erosional, turbulent regime to a depositional, stably stratified regime. Finally, the transition model is solved for a series of case studies and a numerical parameter study is conducted in an attempt to answer the question `when does a turbidity current become extinct?'.

The propulsion mechanics driving the movement of living cells constitutes one of the most incredible engineering works of nature. Active cell motility via the controlled movement of a flagellum beating is among the phylogentically oldest forms of motility, and has been retained in higher level organisms for spermatozoa transport. Despite this ubiquity and importance, the details of how each structural component within the flagellum is orchestrated to generate bending waves, or even the elastic material response from the sperm flagellum, is far from fully understood. By using microbiomechanical modelling and simulation, we develop bio-inspired mathematical models to allow the exploration of sperm motility and the material response of the sperm flagellum. We successfully construct a simple biomathematical model for the human sperm movement by taking into account the sperm cell and its interaction with surrounding fluid, through resistive-force theory, in addition to the geometrically non-linear response of the flagellum elastic structure. When the surrounding fluid is viscous enough, the model predicts that the sperm flagellum may buckle, leading to profound changes in both the waveforms and the swimming cell trajectories. Furthermore, we show that the tapering of the ultrastructural components found in mammalian spermatozoa is essential for sperm migration in high viscosity medium. By reinforcing the flagellum in regions where high tension is expected this flagellar accessory complex is able to prevent tension-driven elastic instabilities that compromise the spermatozoa progressive motility. We equally construct a mathematical model to describe the structural effect of passive link proteins found in flagellar axonemes, providing, for the first time, an explicit mathematical demonstration of the counterbend phenomenon as a generic property of the axoneme, or any cross-linked filament bundle. Furthermore, we analyse the differences between the elastic cross-link shear and pure material shear resistance. We show that pure material shearing effects from Cosserat rod theory or, equivalently, Timoshenko beam theory or are fundamentally different from elastic cross-link induced shear found in filament bundles, such as the axoneme. Finally, we demonstrate that mechanics and modelling can be utilised to evaluate bulk material properties, such as bending stiffness, shear modulus and interfilament sliding resistance from flagellar axonemes its constituent elements, such as microtubules.

This study focuses on the Sun's influence on the motion near the triangular libration points of the Earth-Moon system. It is known that there exists a very strong resonant perturbation near those points that produces large deviations from the libration points, with an amplitude of about 250,000 km and a period of 1,500 days. However, it has been shown that it is possible to find initial conditions that negate the effects of that perturbation, even resulting in stable, although very large, periodic orbits. Using two different models, the goal of this research is to determine the initial configurations of the Earth-Moon-Sun system that produce minimal deviations from the libration points, and to provide a better understanding of the dynamics of this highly nonlinear problem. First, the Bicircular Problem (BCP) is considered, which is an idealized model of the Earth-Moon-Sun System. The impact of the initial configuration of the Earth-Moon-Sun system is studied for various propagation times and it is found that there exist two initial configurations that produce minimal deviations from L₄ or L₅. The resulting trajectories are very sensitive to the initial configuration, as the mean deviation from the libration points can decrease by 30,000 km with less than a degree change in the initial configuration. Two critical initial configurations of the system were identified that could allow a particle to remain within 30,000 km of the libration points for as long as desired. A more realistic model, based on JPL ephemerides, is also used, and the influence of the initial epoch on the motion near the triangular points is studied. Through the year 2007, 51 epochs are found that produce apparently stable librational motion near L₄, and 60 near L₅. But the motion observed depends greatly on the initial epoch. Some epochs are even found to significantly reduce the deviation from L₄ and L₅, with the spacecraft remaining within at most 90,000 km from the triangular points for upwards of 3,000 days. Similarly to what was observed in the BCP, these trajectories are found to be extremely sensitive to the initial epoch. / text