Thesis (Ph.D.)--University of Texas at Arlington, 2009.

The main motivation of this thesis comes from a desire to understand the behaviours of blood flow in the vicinity of atheroma. The initiation and development of atherosclerosis in arteries are normally observed in the areas of low or oscillating wall shear stress, such as on the outer wall of a bifurcation and the inside of the bends. We start by building on the background to the areas related to our models. We focus on the models of fluid flow travelling through a curved tube of uniform curvature because most arteries are tapered and curved. The flow of an incompressible Newtonian fluid in a curved tube is modelled. The derivation of the corresponding equations of the motion is presented. The equations are then solved for a steady and oscillatory driving axial pressure gradient. In each case, the flow is governed by different dimensionless parameters. The problem is solved for a variety of parameter regimes by using asymptotic technique as well as numerical method. Some aspects of thin-film flows are studied. The well-known thin film equation is derived using lubrication theory. The stability of a thin film in a straight tube and the effects of a surfactant droplet on a liquid film are presented. The moving contact line problem, one of the controversial topics in fluid dynamics, is also discussed. The leading-order equations governing thin-film flow over a stationary curved substrate is derived. Various approaches and the application of flow on particular substrates are shown. Finally, we model two-layer viscous fluids using lubrication approximation. By assuming the thickness of a lower liquid layer is much thinner than that of the upper liquid layer, the equation governing the liquid-liquid interface is derived. The steady-state and trasient solutions of the evolution equation is computed both analytically and computationally.

Recently an iterative method was formulated employing a new splitting strategy for the solution of tridiagonal systems of difference equations. The method was successful in solving the systems of equations arising from one dimensional initial boundary value problems, and a theoretical analysis for proving the convergence of the method for systems whose constituent matrices are positive definite was presented by Evans and Sahimi [22]. The method was known as the Alternating Group Explicit (AGE) method and is referred to as AGE-1D. The explicit nature of the method meant that its implementation on parallel machines can be very promising. The method was also extended to solve systems arising from two and three dimensional initial-boundary value problems, but the AGE-2D and AGE-3D algorithms proved to be too demanding in computational cost which largely reduces the advantages of its parallel nature. In this thesis, further theoretical analyses and experimental studies are pursued to establish the convergence and suitability of the AGE-1D method to a wider class of systems arising from univariate and multivariate differential equations with symmetric and non symmetric difference operators. Also the possibility of a Chebyshev acceleration of the AGE-1D algorithm is considered. For two and three dimensional problems it is proposed to couple the use of the AGE-1D algorithm with an ADI scheme or an ADI iterative method in what is called the Explicit Alternating Direction (EAD) method. It is then shown through experimental results that the EAD method retains the parallel features of the AGE method and moreover leads to savings of up to 83 % in the computational cost for solving some of the model problems. The thesis also includes applications of the AGE-1D algorithm and the EAD method to solve some problems of fluid dynamics such as the linearized Shallow Water equations, and the Navier Stokes' equations for the flow in an idealized one dimensional Planetary Boundary Layer. The thesis terminates with conclusions and suggestions for further work together with a comprehensive bibliography and an appendix containing some selected programs.

In this thesis we derive and solve mathematical models for surfactant systems with differing adsorption structures at interfaces. The first part of this thesis considers two dynamic experimental set-ups for which we derive the associated mathematical surfactant–fluid description. Firstly we consider the behaviour of a weakly interacting polymer–surfactant solution under the influence of a steady straining flow. We reduce the model using asymptotic methods to predict the regimes under which we observe phase transitions of the species in the system and show how the bulk dynamics couple to the surfactant adsorption. Secondly we model an experiment to observe the desorption kinetics of a surfactant monolayer, designed to emulate the 'rinse mechanism' used for the removal of surfactant-containing products using water. Through the comparison of our model with experimental data we derive a semi-empirical relationship that describes the variation in depth of a near-surface diffusive boundary layer with the reduced Peclet number. We then employ a combination of asymptotic and numerical techniques that validate this result. The second part of this thesis is concerned with surfactant systems that exhibit more pronounced adsorption at the interface due to the surfactant monomers no longer arranging themselves in a single layer, as is typically the case, but rather in multiple layers. Such self-assembled structures are commonly referred to as multilayers. We derive a simplified model that describes the rearrangement of surfactant within the multilayer structure and draw comparisons between the features of our model and experimental observations. We consider an extension of the theory to the situation of multilayer formation between two adsorbing interfaces, which is governed by an implicit free-boundary problem. We also consider incorporation of bulk solution effects, such as the addition of an electrolyte. Finally, we draw our conclusions and suggest further theoretical and experimental work related to the models presented in this thesis.

This thesis concerns the modelling of sound propagation in an urban environment. For most of the thesis a point source of sound source is assumed, and both 2D and 3D geometries are considered. Buildings are modelled as rigid blocks, with the effects of surface inhomogeneities neglected. In the time-harmonic case, assuming that the wavelength is short compared to typical lengthscales of the domain (street widths and lengths), ray theory is used to derive estimates for the time-averaged acoustic power flows through a network of interconnecting streets in the form of integrals over ray angles. In the impulsive case, the propagation of wave-field singularities in the presence of obstacles is considered, and a general principle concerning the weakening of singularities when they are diffracted by edges and vertices is proposed. The problem of switching on a time-harmonic source is also studied, and an exact solution for the diffraction of a switched on plane wave by a rigid half-line is obtained and analysed. The pulse diffraction theory is then applied in a study of the inverse problem for an impulsive source, where the aim is to locate an unknown source using Time Differences Of Arrival (TDOA) at multiple receivers. By using reflected and diffracted pulse arrivals, the standard free-space TDOA method is extended to urban environments. In particular, approximate source localisation is found to be possible even when the exact building distribution is unknown.

This thesis is concerned with the modelling of miscible fluid flow through porous media, with the intended application being the displacement of oil from a reservoir by a solvent with which the oil is miscible. The primary difficulty that we encounter with such modelling is the existence of a fingering instability that arises from the viscosity and the density differences between the oil and solvent. We take as our basic model the Peaceman model, which we derive from first principles as the combination of Darcy’s law with the mass transport of solvent by advection and hydrodynamic dispersion. In the oil industry, advection is usually dominant, so that the Péclet number, Pe, is large. We begin by neglecting the effect of density differences between the two fluids and concentrate only on the viscous fingering instability. A stability analysis and numerical simulations are used to show that the wavelength of the instability is proportional to Pe^−1/2, and hence that a large number of fingers will be formed. We next apply homogenisation theory to investigate the evolution of the average concentration of solvent when the mean flow is one-dimensional, and discuss the rationale behind the Koval model. We then attempt to explain why the mixing zone in which fingering is present grows at the observed rate, which is different from that predicted by a naive version of the Koval model. We associate the shocks that appear in our homogenised model with the tips and roots of the fingers, the tip-regions being modelled by Saffman-Taylor finger solutions. We then extend our model to consider flow through porous media that are heterogeneous at the macroscopic scale, and where the mean flow is not one dimensional. We compare our model with that of Todd & Longstaff and also models for immiscible flow through porous media. Finally, we extend our work to consider miscible displacements in which both density and viscosity differences between the two fluids are relevant.

Understanding the perfusion of blood and drugs in tumours is fundamental to foreseeing the efficacy of treatment regimes and predicting tumour growth. In particular, the dependence of these processes on the tumour vascular structure is poorly established. The objective of this thesis is to derive effective equations describing blood, and drug perfusion in vascular tumours, and specifically to determine the dependence of these on the tumour vascular structure. This dependence occurs through the interaction between two different length scales - that which characterizes the structure of the vascular network, and that which characterizes the tumour as a whole. Our method throughout is to use homogenization as a tool to evaluate this interaction. In Chapter 1 we introduce the problem. In Chapter 2 we develop a theoretical model to describe fluid flow in solid tumours through both the vasculature and the interstitium, at a number of length scales. Ultimately we homogenize over a network of capillaries to form a coupled porous medium model in terms of a vascular density. Whereas in Chapter 2 it is necessary to specify the vascular structure to derive the effective equations, in Chapter 3 we employ asymptotic homogenization through multiple scales to derive the coupled equations for an arbitrary periodic vascular network. In Chapter 4, we extend this analysis to account for advective and diffusive transport of anticancer drugs delivered intravenously; we consider a range of reaction properties in the interstitium and boundary conditions on the vascular wall. The models derived in Chapters 2–4 could be applied to any drug type and treatment regime; to demonstrate their potential, we simulate the delivery of vinblastine in dorsal skinfold chambers in Chapter 5 and make quantitative predictions regarding the optimal treatment regime. In the final Chapter we summarize the main results and indicate directions for further work.

Motility of flagellated bacteria has been a topic of increasing scientific interest over the past decades, attracting the attention of mathematicians, physicists, biologists and engineers alike. Bacteria and other micro-organisms cause substantial damage through biofilm growth on submerged interfaces in water cooling systems, ship hulls and medical implants. This gives social and economic motivations for learning about how micro-organisms swim and behave in different environments. Fluid flows on such small scales are dominated by viscosity and therefore behave differently from the inertia-dominated flows that we are more familiar with, making bacterial motility a physically intriguing phenomenon to study as well. We use the boundary element method (BEM) to simulate the motion of singly flagellated bacteria in a viscous, Newtonian fluid. One of our main objectives is to investigate the influence of external surfaces on swimming behaviour. We show that the precise shape of the cell body and flagellum can be important for determining boundary behaviour, in particular, whether bacteria are attracted or repelled from surfaces. Furthermore, we investigate the types of motion that may arise between two parallel plates and in rectangular channels of fluid and show how these relate to the plane boundary interactions. As an extension to original models of flagellar propulsion in bacteria that assume a rotation of the rigid helical flagellum about an axis fixed relative to the cell body, we consider flexibility of the bacterial hook connecting the aforementioned parts of the swimmer. This is motivated by evidence that the hook is much more flexible than the rest of the flagellum, which we therefore treat as a rigid structure. Elastic dynamics of the hook are modelled using the equations for a Kirchhoff rod. In some regimes, the dynamics are well described by a rigid hook model but we find the possibility of additional modes of behaviour.

Ice sheets are among the key controls on global climate and sea level. A detailed understanding of their dynamics is crucial to make accurate predictions of their future mass balance. Ice streams are the dominant negative component in this balance, accounting for up to 90% of the Antarctic ice flux into ice shelves and ultimately into the sea. Despite their importance, our understanding of ice-stream dynamics is far from complete. A range of observations associate ice streams with meltwater. Meltwater lubricates the ice at its bed, allowing it to slide with less internal deformation. It is believed that ice streams may appear due to a localisation feedback between ice flow, basal melting and water pressure in the underlying sediments. This thesis aims to address the instability of ice-stream formation by considering potential feedbacks between the basal boundary and ice flow. Chapter 2 considers ice-flow models, formulating a model that is capable of capturing the leading-order dynamics of both a slow-moving ice sheet and rapidly flowing ice streams. Chapter 3 investigates the consequences of applying different phenomenological sliding laws as the basal boundary condition in this ice-flow model. Chapter 4 presents a model of subglacial water flow below ice sheets, and particularly below ice streams. This provides a more physical representation of processes occurring at the bed. Chapter 5 then investigates the coupled behaviour of the water with the sediment, and Chapter 6 the coupled behaviour of the water with the ice flow. Under some conditions this coupled system gives rise to ice streams due to instability of the internal dynamics.

In this thesis, we derive, extend and generalise various aspects of impact theory and splash dynamics. Our methods throughout will involve isolating small parameters in our models, which we can utilise using the language of matched asymptotics. In Chapter 1 we briefly motivate the field of impact theory and outline the structure of the thesis. In Chapter 2, we give a detailed review of classical small-deadrise water entry, Wagner theory, in both two and three dimensions, highlighting the key results that we will use in our extensions of the theory. We study oblique water entry in Chapter 3, in which we use a novel transformation to relate an oblique impact with its normal-impact counterpart. This allows us to derive a wide range of solutions to both two- and three-dimensional oblique impacts, as well as discuss the limitations and breakdown of Wagner theory. We return to vertical water-entry in Chapter 4, but introduce the air layer trapped between the impacting body and the liquid it is entering. We extend the classical theory to include this air layer and in the limit in which the density ratio between the air and liquid is sufficiently small, we derive the first-order correction to the Wagner solution due to the presence of the surrounding air. The model is presented in both two dimensions and axisymmetric geometries. In Chapter 5 we move away from Wagner theory and systematically derive a series of splash jet models in order to find possible mechanisms for phenomena seen in droplet impact and droplet spreading experiments. Our canonical model is a thin jet of liquid shot over a substrate with a thin air layer trapped between the jet and the substrate. We consider a variety of parameter regimes and investigate the stability of the jet in each regime. We then use this model as part of a growing-jet problem, in which we attempt to include effects due to the jet tip. In the final chapter we summarise the main results of the thesis and outline directions for future work.