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Dissertations / Theses on the topic 'Fluid mechanics – Mathematics'
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1
Smith, Andrew. "The fluid mechanics of embryonic nodal cilia." Thesis, University of Birmingham, 2013. http://etheses.bham.ac.uk//id/eprint/4626/.
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Symmetry breaking of the left-right body axis is a crucial step in development for many vertebrate species. In many this is initiated with a directional cilia-driven fluid flow in the organising structure. This work focuses on the mouse and the zebrafish organising structures, the node and Kupffer's vesicle, wherein cilia perform a tilted rotation producing an asymmetric flow. Using singularities of Stokes flow, slender body theory and the boundary integral equation, a computational model of flow in the mouse node for a range of cilia configurations simulating developmental stages is developed and run on the University of Birmingham's cluster, BlueBEAR. The results show the emergence of a directional flow as the cilia tilt increases. To model the Kupffer's vesicle the regularised boundary integral equation is used with a mesh representation of the entire domain to investigate potential cilia mechanisms that produce the observed flow as there is not a consensus. The results show that a combination of the experimental observations could be a sufficient mechanism. This model is expanded using observations of cilia with two rotation frequencies which are incorporated by allowing such cilia to ‘wobble’. This wobble accentuates the asymmetric flow in wildtype embryos and diminishes it in mutant embryos. All of these results agree well with experiment suggesting that vertebrates develop a combination of rotation mechanisms in their organising structures before an appropriate symmetry breaking flow is established.
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Dashti, Masoumeh. "Some problems in the mathematical theory of fluid mechanics." Thesis, University of Warwick, 2008. http://wrap.warwick.ac.uk/3665/.
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This thesis addresses three problems related to the mathematical theory of fluid mechanics. Firstly, we consider the three-dimensional incompressible Navier-Stokes equations with an initial condition that has H1-Sobolev regularity. We show that there is an a posteriori condition that, if satisfied by the numerical solutions of the equations, guarantees the existence of a strong solution and therefore the validity of the numerical computations. This is an extension of a similar result proved by Chernyshenko, Constantin, Robinson & Titi (2007) to less regular solutions not considered by them. In the second part, we give a simple proof of uniqueness of fluid particle trajectories corresponding to the solution of the d-dimensional Navier Stokes equations, d = 2, 3, with an initial condition that has H(d/2)−1-Sobolev regularity. This result has been proved by Chemin & Lerner (1995) using the Littlewood-Payley theory for the flow in the whole space Rd. We provide a significantly simpler proof, based on the decay of Sobolev norms ( of order more than (d/2)−1 ) of the velocity field after the initial time, that is also valid for the more physically relevant case of bounded domains. The last problem we study is the motion of a fluid-rigid disk system in the whole plane at the zero limit of the rigid body radius. We consider one rigid disk moving with the fluid flow and show that when the radius of the disk goes to zero, the solution of this system converges, in an appropriate sense, to the solution of the Navier-Stokes equations describing the motion of only fluid in the whole plane. We then prove that the trajectory of the centre of the disk, at the zero limit of its radius, coincides with a fluid particle trajectory. We also show an equivalent result for the limiting motion of a spherical tracer in R3, over a small enough time interval.
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Breward, C. J. W. "The mathematics of foam." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.300849.
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The aim of this thesis is to derive and solve mathematical models for the flow of liquid in a foam. A primary concern is to investigate how so-called `Marangoni stresses' (i.e. surface tension gradients), generated for example by the presence of a surfactant, act to stabilise a foam. We aim to provide the key microscopic components for future foam modelling. We begin by describing in detail the influence of surface tension gradients on a general liquid flow, and various physical mechanisms which can give rise to such gradients. We apply the models thus devised to an experimental configuration designed to investigate Marangoni effects. Next we turn our attention to the flow in the thin liquid films (`lamellae') which make up a foam. Our methodology is to simplify the field equations (e.g. the Navier-Stokes equations for the liquid) and free surface conditions using systematic asymptotic methods. The models so derived explain the `stiffening' effect of surfactants at free surfaces, which extends considerably the lifetime of a foam. Finally, we look at the macroscopic behaviour of foam using an ad-hoc averaging of the thin film models.
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Cosper, Lane. "Existence of a Unique Solution to a System of Equations Modeling Compressible Fluid Flow with Capillary Stress Effects." Thesis, Texas A&M University - Corpus Christi, 2018. http://pqdtopen.proquest.com/#viewpdf?dispub=10790012.
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<p> The purpose of this thesis is to prove the existence of a unique solution to a system of partial differential equations which models the flow of a compressible barotropic fluid under periodic boundary conditions. The equations come from modifying the compressible Navier-Stokes equations. The proof utilizes the method of successive approximations. We will define an iteration scheme based on solving a linearized version of the equations. Then convergence of the sequence of approximate solutions to a unique solution of the nonlinear system will be proven. The main new result of this thesis is that the density data is at a given point in the spatial domain over a time interval instead of an initial density over the entire spatial domain. Further applications of the mathematical model are fluid flow problems where the data such as concentration of a solute or temperature of the fluid is known at a given point. Future research could use boundary conditions which are not periodic.</p><p>
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Rahantamialisoa, Faniry Nadia Zazaravaka. "Complex fluid dynamical computations via the Finite Volume Method." Master's thesis, University of Cape Town, 2018. http://hdl.handle.net/11427/29860.
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Numerical simulations of the complex flows of viscoelastic fluids are investigated. The viscoelastic fluids are modelled, primarily, via the Johnson-Segalman constitutive model. Our Numerical approach is based on finite volume method, based on the Johnson-Segalman constitutive model and implemented on the OpenFOAM® platform. The Johnson-Segalman model also easily reduces to the Oldroyd-B model under certain conditions of the material parameters. Since computations using the Oldroyd-B model have been extensively documented in the literature, we take advantage of the mathematical modelling connection between the Johnson-Segalman and Oldroyd-B models to validate the accuracy of our Johnson-Segalman solver via reduction to the Oldroyd-B model. Numerical validation of our results is conducted via the most commonly used benchmark problems. The final aim of our work is to assess the viability and efficiency of our numerical solver via an investigation into the complex fluid dynamical processes associated with shear banding.
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Fay, Gemma Louise. "Mathematical modelling of turbidity currents." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:62bb9382-1c50-47f3-8f59-66924cc31760.
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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?'.
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Garratt, T. J. "The numerical detection of Hopf bifurcations in large systems arising in fluid mechanics." Thesis, University of Bath, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.292839.
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Chutsagulprom, Nawinda. "Thin film flows in curved tubes." Thesis, University of Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:35071002-e487-4cd3-b85f-3aca6dcb0c93.
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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.
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Shipley, Rebecca Julia. "Multiscale modelling of fluid and drug transport in vascular tumours." Thesis, University of Oxford, 2009. http://ora.ox.ac.uk/objects/uuid:8f663f70-8d23-49ad-8348-1763359d8f62.
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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.
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Al-Wali, Azzam Ahmad. "Explicit alternating direction methods for problems in fluid dynamics." Thesis, Loughborough University, 1994. https://dspace.lboro.ac.uk/2134/6840.
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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.
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Cheng, Yi-fen. "The structure of shock waves in an asymptotic magnetohydrodynamics system." Diss., The University of Arizona, 1993. http://hdl.handle.net/10150/186343.
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We study an asymptotic MHD model system. In particular, we show its proximity to MHD system by studying the fundamental properties of MHD system in our model system. We prove the existence and boundness of the structures of intermediate shock waves in the planar model system and in the non-planar model system, respectively. We also extend the Liu's theorem on the nonlinear instability of the travelling wave solutions of the Derivative Schroedinger equation to our more general model.
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Moore, Matthew Richard. "New mathematical models for splash dynamics." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:c94ff7f2-296a-4f13-b04b-e9696eda9047.
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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.
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Morgan, Cara Ellen. "Mathematical modelling of surfactant adsorption structures at interfaces." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:9bbe7487-6d53-4046-b0b8-78a156628130.
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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.
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Benham, Graham P. "Mathematical modelling and optimisation of Venturi-enhanced hydropower." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:1efacb88-d5e5-450d-83a7-d7fe0873b6fb.
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In this thesis we study a novel type of hydropower generation which uses a Venturi contraction to amplify the pressure drop across a turbine, allow- ing for cost-effective hydropower generation in situations where the head drop is small, such as in rivers and weirs. The efficiency is sensitive to how the secondary flow, which passes through the turbine, mixes with the accelerated primary flow, which is diverted around the turbine, within the confines of a closed geometry. In particular, it is important to understand the behaviour of the turbulent shear layers between the primary and sec- ondary flows, which grow downstream, mixing the flows together. The behaviour of the shear layers in the expanding part of the Venturi con- traction is strongly dependent on the shape of the channel. An important consequence of the channel shape, and hence the flow behaviour, is the degree of pressure amplification across the turbine, which determines the amount of generated power. We focus on mathematically modelling the mixing of the flows in turbu- lent shear layers, and we investigate two different ways to increase pres- sure amplification: optimising the shape of the channel, and using swirl to enhance mixing. The channel shape optimisation reveals an interest- ing balance between the effects of mixing and wall drag. Wide channel expansion tends to accentuate non-uniform flow, causing poor pressure amplification, whilst shallow expansion creates enhanced wall drag, which is also detrimental to pressure amplification. We show how the maximum power is generated with a channel shape that strikes a balance between these two effects. We find that swirl enhances mixing by increasing shear layer growth rates, but it produces large pressure losses in doing so, and for large amounts of swirl a slowly recirculating region can form along the channel centreline. Whilst swirl does not improve efficiency, there may be some inevitable swirl present in the flow, and we show how this affects the optimum channel shape. We also establish the criteria for the existence of such a recirculation region so that it may be avoided.
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Eisenträger, Almut. "Finite element simulation of a poroelastic model of the CSF system in the human brain during an infusion test." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:372f291f-cf36-48ef-8ce8-d4c102bce9e3.
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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.
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Xu, Xiong. "Numerical analysis of blood flow in 3-D arterial bifurcations." Thesis, City University London, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.316023.
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Li, Siran. "Analysis of several non-linear PDEs in fluid mechanics and differential geometry." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:20866cbb-e5ab-4a6b-b9dc-88a247d15572.
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In the thesis we investigate two problems on Partial Differential Equations (PDEs) in differential geometry and fluid mechanics. First, we prove the weak L<sup> p</sup> continuity of the Gauss-Codazzi-Ricci (GCR) equations, which serve as a compatibility condition for the isometric immersions of Riemannian and semi-Riemannian manifolds. Our arguments, based on the generalised compensated compactness theorems established via functional and micro-local analytic methods, are intrinsic and global. Second, we prove the vanishing viscosity limit of an incompressible fluid in three-dimensional smooth, curved domains, with the kinematic and Navier boundary conditions. It is shown that the strong solution of the Navier-Stokes equation in H<sup> r+1</sup> (r > 5/2) converges to the strong solution of the Euler equation with the kinematic boundary condition in H<sup> r</sup>, as the viscosity tends to zero. For the proof, we derive energy estimates using the special geometric structure of the Navier boundary conditions; in particular, the second fundamental form of the fluid boundary and the vorticity thereon play a crucial role. In these projects we emphasise the linkages between the techniques in differential geometry and mathematical hydrodynamics.
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Warneford, Emma S. "The thermal shallow water equations, their quasi-geostrophic limit, and equatorial super-rotation in Jovian atmospheres." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:6604fcac-afe6-4abe-8a6f-6a09de4f933f.
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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.
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Reid, Francis John Edward School of Mathematics UNSW. "The weakly nonlinear stability of an oscillatory fluid flow." Awarded by:University of New South Wales. School of Mathematics, 2006. http://handle.unsw.edu.au/1959.4/33364.
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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.
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Curtis, Mark Peter. "Aspects of low Reynolds number microswimming using singularity methods." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:13dcb39b-f5b7-4d46-92d4-21a9afbecd08.
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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.
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Dalwadi, Mohit. "Flow and nutrient transport problems in rotating bioreactor systems." Thesis, University of Oxford, 2014. https://ora.ox.ac.uk/objects/uuid:1d7298b7-cdf5-4240-a79c-b7b69f662c1a.
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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.
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Hewett, David Peter. "Sound propagation in an urban environment." Thesis, University of Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:e7a1d40b-2bf4-4f48-8a6b-ce6f575e955e.
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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.
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Gadelha, Hermes. "Mathematical modelling of human sperm motility." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:34a11669-5d14-470b-b10b-361cf3688a30.
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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.
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Sutton, Kaylee B. "Surface Nonuniformities in Waterborne Coatings due to Evaporative Mechanisms." University of Akron / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=akron1470407446.
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Tully, Brett. "Allostasis of cerebral water : modelling the transport of cerebrospinal fluid." Thesis, University of Oxford, 2010. http://ora.ox.ac.uk/objects/uuid:168586f0-f34a-4d5e-8acf-822cd0e1bfe2.
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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.
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Williams, Paul David. "Nonlinear interactions of fast and slow modes in rotating, stratified fluid flows." Thesis, University of Oxford, 2003. http://ora.ox.ac.uk/objects/uuid:5365c658-ab60-41e9-b07b-0f635909835e.
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This thesis describes a combined model and laboratory investigation of the generation and mutual interactions of fluid waves whose characteristic scales differ by an order of magnitude or more. The principal aims are to study how waves on one scale can generate waves on another, much shorter scale, and to examine the subsequent nonlinear feedback of the short waves on the long waves. The underlying motive is to better understand such interactions in rotating, stratified, planetary fluids such as atmospheres and oceans. The first part of the thesis describes a laboratory investigation using a rotating, two-layer annulus, forced by imposing a shear across the interface between the layers. A method is developed for making measurements of the two-dimensional interface height field which are very highly-resolved both in space and time. The system's linear normal modes fall into two distinct classes: 'slow' waves which are relatively long in wavelength and intrinsic period, and 'fast' waves which are much shorter and more quickly-evolving. Experiments are performed to categorize the flow at a wide range of points in the system's parameter space. At very small background rotation rates, the interface is completely devoid of waves of both types. At higher rates, fast modes only are generated, and are shown to be consistent with the Kelvin-Helmholtz instability mechanism based on a critical Richardson number. At rotation rates which are higher still, baroclinic instability gives rise to the onset of slow modes, with subsequent localized generation of fast modes superimposed in the troughs of the slow waves. In order to examine the generation mechanism of these coexisting fast modes, and to assess the extent of their impact upon the evolution of the slow modes, a quasi-geostrophic numerical model of the laboratory annulus is developed in the second part of the thesis. Fast modes are filtered out of the model by construction, as the phase space trajectory is confined to the slow manifold, but the slow wave dynamics is accurately captured. Model velocity fields are used to diagnose a number of fast wave radiation indicators. In contrast to the case of isolated fast waves, the Richardson number is a poor indicator of the generation of the coexisting fast waves that are observed in the laboratory, and so it is inferred that these are not Kelvin-Helmholtz waves. The best indicator is one associated with the spontaneous emission of inertia-gravity waves, a generalization of geostrophic adjustment radiation. A comparison is carried out between the equilibrated wavenumbers, phase speeds and amplitudes of slow waves in the laboratory (which coexist with fast modes), and slow waves in the model (which exist alone). There are significant differences between these wave properties, but it is shown that these discrepancies can be attributed to uncertainties in fluid properties, and to model approximations apart from the neglect of fast modes. The impact of the fast modes on the slow modes is therefore sufficiently small to evade illumination by this method of inquiry. As a stronger test of the interaction, a stochastic parameterization of the inertia-gravity waves is included in the model. Consistent with the laboratory/model intercomparison, the parameterized fast waves generally have only a small impact upon the slow waves. However, sufficiently close to a transition curve between two different slow modes in the system's parameter space, it is shown that the fast modes can exert a dominant influence. In particular, the fast modes can force spontaneous transitions from one slow mode to another, due to the phenomenon of stochastic resonance. This finding should be of interest to the meteorological and climate modelling communities, because of its potential to affect model reliability.
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Wren, Elisabeth Mary Katie. "Geometric effects on phase split at a large diameter T-junction." Thesis, University of Nottingham, 2001. http://eprints.nottingham.ac.uk/11082/.
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The separation of gas-liquid flows is a necessary part of many industrial processes. Thus, it has received much attention over the years with the ultimate aim of reducing equipment costs whilst maintaining or improving efficiency. Traditionally, the Petroleum Industry has relied heavily on conventional vessel separators which are bulky, expensive and have a high inventory. Research has indicated that a cheap alternative may be a simple pipe junction. It has been shown that gas-liquid flows can be divided at pipe junctions in such a manner that there is a partial separation of the phases. The result is two streams - one richer in gas than the initial feed and the other richer in liquid. If the phases can be separated, albeit partially, at a simple pipe junction then the need for a large separator is diminished. Within this thesis the use of a simple T-junction is considered as a continuous, compact, economical partial phase separator with a minimal inventory for use within the oil industry. The main objective was to gain a better understanding of how a gas-liquid flow is divided at a large diameter T-junction and how the flow split is affected by T-junction geometry. Firstly, the orientation of the side arm from the horizontal was considered with both a regular (inlet arm diameter == branch arm diameter = 0.127m) T-junction and a reduced (branch arm to inlet diameter ratio = 0.6) T-junction. The side arm was placed horizontally (0'), vertically upwards (+90') and vertically downwards (-90') and the phase split of air water annular and stratified flows were investigated. To improve the phase separation characteristics of the regular T-junction, inserts protruding from the side arm into the main pipe were considered and for the junction with a vertically downwards side arm a U-bend was used to reduce the fraction of gas pulled through. The experimental investigation was expanded to incorporate the effect of placing two regular T-junctions in series. With the branch arm of the first placed vertically upwards (+90'). and the second vertically downwards (-90') a pure gas stream and a liquid rich stream were created from the multi-phase inlet. Reducing the sidearm diameter of the second junction lowered the fraction of gas drawn off in the liquid rich stream. The physical separation distance the T-junctions was found to have little effect on phase split. The interaction of the two junctions are interdependent and the phase split results from the two junction system was found to be more complex than simply considering the results of two individual T-junctions. Being able to predict the phase split at a junction is vital if they are to be considered seriously within industrial settings. The case of a regular T-junction with a vertically downwards (-90') side arm has received little specific attention. From the linear nature of the phase split results it was determined that if two key points could be accurately predicted then the phase split results can be determined. The "onset of gas take off', the fraction of liquid diverted down the branch arm when the first fraction of gas is pulled through, was successfully related to the bubble rise velocity of the gas entrained in the liquid column trapped in the branch arm. The "critical gas take off”, the fraction of gas diverted when all the liquid is drawn down the branch arm, was determined by relating the fluid flow to the motion of a failing particle.
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Kirkegaard, Julius Bier. "Physical and stochastic aspects of microorganism behaviour." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/277543.
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This thesis studies physical and stochastic aspects of microorganisms. From the point of view of $\textit{physics}$, the studies in this thesis are motivated by the goal of gaining biological insight using the machinery of physics and mathematics. From the point of view of $\textit{biology}$, the studies in this thesis focus primarily on choanoflagellates, eukaryotes that are the closest living unicellular relatives of animals. This choice of model organism was motivated by the important biological question of the origin of multicellularity. Why was it that single-celled organisms evolved to become multicellular? In particular, we study closely the species $\textit{Salpingoeca rosetta}$, which has the ability to form colonies that resemble true multicellular organisms. A large part of this thesis deals with the random walks of microorganisms. We study these active random walks both for single cells and those composed of individual organisms adhered together. The latter colonial random walkers are typified by choanoflagellates. We develop quantitative theories and use these to extract physical parameters. The increasing ocean oxygen levels in the Precambrian era are thought to be an important factor in the emergence of complex multicellular, animal life. As a first step, we address this situation by studying the response of $\textit{S. rosetta}$ to oxygen gradients. We find that $\textit{S. rosetta}$ displays positive aerotaxis. Analysis of the spatial population distributions provides evidence for logarithmic sensing of oxygen, which enhances sensing in low oxygen neighbourhoods. Analysis of search strategy models on the experimental colony trajectories finds that choanoflagellate aerotaxis is consistent with stochastic navigation, the statistics of which are captured using an effective continuous version of classical run-and-tumble chemotaxis. We compare this continuous run-to-tumble with the run-and-tumble seen in bacteria by formulating a general model for persistent run-and-tumble. We find that although an optimal persistence does exist for a given tumble frequency, in the full parameter space there is a continuum of optimal solutions. We develop this model further by introducing finite tumble times. Efficient uptake of prey and nutrients from the environment is an important component in the fitness of all microorganisms, and its dependence on size may reveal clues to the origins of evolutionary transitions to multicellularity. We examine these issues in depth for choanoflagellates, finding that in the absence of other requirements and in a homogeneously nutritious environment, the optimal strategy to maximise filter feeding is to swim fast which favours swimming unicells. In contrast, in large external flows, a sessile form becomes advantageous. Effects of prey diffusion are discussed and are also found to be advantageous for the swimming unicell. Finally, we consider the switching between synchronous and anti-synchronous beating of flagella in the green alga $\textit{Chlamydomonas}$, a phenomenon that results in run-and-tumble behaviour in eukaryotes. We develop a theoretical model to describe this beating and use it to argue that the synchrony itself is obtained intracellularly, whereas the flagella shapes are most likely strongly influenced by hydrodynamic interactions.
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Stewart, Andrew L. "The role of the complete Coriolis force in cross-equatorial transport of abyssal ocean currents." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:6bf3faff-ec7e-4d11-bdfe-c54ae9d03895.
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In studies of the ocean it has become conventional to retain only the component of the Coriolis force associated with the radial component of the Earth’s rotation vector, the so-called “traditional approximation”. We investigate the role of the “non-traditional” component of the Coriolis force, corresponding to the non-radial component of the rotation vector, in transporting abyssal waters across the equator. We first derive a non-traditional generalisation of the multi-layer shallow water equations, which describe the flow of multiple superposed layers of inviscid, incompressible fluid with constant densities over prescribed topography in a rotating frame. We derive these equations both by averaging the three-dimensional governing equations over each layer, and via Hamilton’s principle. The latter derivation guarantees that conservation laws for mass, momentum, energy and potential vorticity are preserved. Within geophysically realistic parameters, including the complete Coriolis force modifies the domain of hyperbolicity of the multi-layer equations by no more than 5%. By contrast, long linear plane waves exhibit dramatic structural changes due to reconnection of the surface and internal wave modes in the long-wave limit. We use our non-traditional shallow water equations as an idealised model of an abyssal current flowing beneath a less dense upper ocean. We focus on the Antarctic Bottom Water, which crosses the equator in the western Atlantic ocean, where the bathymetry forms an almost-westward channel. Cross-equatorial flow is strongly constrained by potential vorticity conservation, which requires fluid to acquire a large relative vorticity in order to move between hemispheres. Including the complete Coriolis force accounts for the fact that fluid crossing the equator in an eastward/westward channel experiences a smaller change in angular momentum, and therefore acquires less relative vorticity. Our analytical and numerical solutions for shallow water flow over idealised channel topography show that the non-traditional component of the Coriolis force facilitates cross-equatorial flow through an almost-westward channel.
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Kimpton, Laura Saranne. "On two-phase flow models for cell motility." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:93c3cc12-4aac-424d-83bf-3e695efb49fe.
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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.
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Hägglund, Jesper. "Simulated cerebrospinal fluid motion due to pulsatile arterial flow : Master Thesis Project." Thesis, Umeå universitet, Institutionen för fysik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-182508.
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All organs, including the brain, need a pathway to remove neurotoxic extracellular proteins. In the brain this is called the glymphatic system. The glymphatic system works by exchanging proteins from interstitial fluids to cerebrospinal fluids. The extracellular proteins are then removed through the cerebrospinal fluid drains. The glymphatic system is believed to be driven by arterial pulsatility, cerebrospinal fluid production and respiration. Cerebrospinal fluids enters the brain alongside arteries. In this project, we investigate if a simulated pulsatile flow in a common carotid artery can drive cerebrospinal fluid flow running along the artery, using computational simulations of a linearly elastic and fluid-structure multiphysical model in COMSOL. Our simulations show that a heartbeat pulse increases the arterial radius of the common carotid artery by 6 %. Experimental data, assessed using 4D magnetic resonance imaging of a living human, show an increase of 13 %. Moreover, our results indicate that arterial displacement itself is not able to drive cerebrospinal fluid flow. Instead, it seems to create a back and forth flow that possibly could help with the protein exchange between the cerebrospinal and interstitial fluids. Overall, the results indicate that the COMSOL Multiphysics linearly elastic model is not ideal for approximations of soft non-linearly elastic solids, such as soft polydimethylsiloxane and artery walls work for stiffer materials. The long term aim is to simulate a part of the glymphatic system and the present work is a starting point to reach this goal. As the simulations in this work are simplified there are more things to test in the future. For example, using the same geometries a non-linear elastic model could be tested. The pulsatile waveform or the geometry could be made more complex. Furthermore the model could be scaled down to represent a penetrating artery in the brain instead of the common carotid artery.
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Booth, Richard J. S. "Miscible flow through porous media." Thesis, University of Oxford, 2008. http://ora.ox.ac.uk/objects/uuid:542d3ec1-2894-4a34-9b93-94bc639720c9.
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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.
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Shum, Henry. "Simulations and modelling of bacterial flagellar propulsion." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:c9f002d8-2939-4744-987e-9a4e659d93ef.
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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.
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Redmon, Jessica. "Stochastic Bubble Formation and Behavior in Non-Newtonian Fluids." Case Western Reserve University School of Graduate Studies / OhioLINK, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=case15602738261697.
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Pearson, Natalie Clare. "Mathematical modelling of flow and transport phenomena in tissue engineering." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:43688cc7-b523-4676-8c41-72db7fc07814.
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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.
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Linders, Viktor. "Error analysis of summation-by-parts formulations : Dispersion, transmission and accuracy." Doctoral thesis, Linköpings universitet, Beräkningsmatematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-143059.
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In this thesis we consider errors arising from finite difference operators on summation-by-parts (SBP) form, used in the discretisation of partial differential equations. The SBP operators are augmented with simultaneous-approximation-terms (SATs) to weakly impose boundary conditions. The SBP-SAT framework combines high order of accuracy with a systematic construction of provably stable boundary procedures, which renders it suitable for a wide range of problems. The first part of the thesis treats wave propagation problems discretised using SBP operators on coarse grids. Unless special care is taken, inaccurate approximations of the underlying dispersion relation materialises in the form of an incorrect propagation speed. We present a procedure for constructing SBP operators with minimal dispersion error. Experiments indicate that they outperform higher order non-optimal SBP operators for flow problems involving high frequencies and long simulation times. In the second part of the thesis, the formal order of accuracy of SBP operators near boundaries is analysed. We prove that the order in the interior of a diagonal norm based SBP operator must be at least twice that of the boundary stencil, irrespective of the grid point distribution near the boundary. This generalises the classical theory posed on uniform and conforming grids. We further show that for a common class of SBP operators, the diagonal norm defines a quadrature rule of the same order as the interior stencil. Again, this result is independent of the grid. In the final contribution if the thesis, we introduce the notion of a transmission problem to describe a general class of problems where different dynamics are coupled in time. Well-posedness and stability analyses are performed for continuous and discrete problems. A general condition is obtained that is necessary and sufficient for the transmission problem to satisfy an energy estimate. The theory provides insights into the coupling of fluid flow models, multi-block formulations, numerical filters, interpolation and multi-grid implementations.
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Congedo, Pietro Marco. "Contributions to the reliability of numerical simulations in fluid mechanics. Application to the flow simulation of thermodynamically complex gases." Habilitation à diriger des recherches, Université Sciences et Technologies - Bordeaux I, 2013. http://tel.archives-ouvertes.fr/tel-00940088.
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At the interface of physics, mathematics, and computer science, Uncertainty Quanti cation (UQ) aims at developing a more rigorous framework and more reliable methods to characterize the impact of uncertainties on the prediction of Quantities Of Interest (QOI). Despite signi cant improvements done in the last years in UQ methods for Fluid Mechanics, there is nonetheless a long way to go before there can be talk of an accurate prediction when considering all the numerous sources of uncertainties of the physical problem (boundary conditions, physical models, geometric tolerances, etc), in particular for shock-dominated problems. This manuscript illustrates my main contributions for improving the reliability of the numerical simulation in Fluid Mechanics: i) the development of e cient and exible schemes for solving at low-cost stochastic partial di erential equations for compressible ows, ii) various works concerning variancebased and high-order analysis, iii) the design of some low-cost techniques for the optimization under uncertainty. The application of interest is the robust design of turbines for Organic Rankine Cycles (ORC). Some contributions to the numerical ow prediction of the thermodynamically complex gases involved in ORC will be presented. This manuscript is divided in two parts. In the rst part, some intrusive algorithms are introduced that feature an innovative formulation allowing the treatment of discontinuities propagating in the coupled physical/stochastic space for shock-dominated compressible ows. Then, variance and higher-order based decompositions are described, that could alleviate problems with large number of uncertainties by performing a dimension reduction with an improved control. Some ANOVAbased analyses are also applied to several ows displaying various types of modeling uncertainties, be it cavitation, thermodynamic or turbulence modeling. Two algorithms for handling stochastic inverse problems are then introduced for improving input uncertainty characterization by directly using experimental data. Finally, robust-optimization algorithms are introduced, that are e cient when dealing with a large number of uncertainties, relying on di erent formulations, i.e. with decoupled/ coupled approaches between the stochastic and the optimization solvers. The second part is devoted to the study of dense gas ow in ORC-cycles, which represent a highly demanding eld of application as far as ow simulation reliability is concerned. The numerical ingredients necessary for this kind of simulation are described. Then, some recent results are illustrated : i) high- delity turbine computations; ii) a feasibility study concerning the appearance and the occurrence of a Rarefaction Shock Wave, using experimental data and di erent operating conditions (in monophasic and two-phase ows); iii) a stochastic study concerning the thermodynamic model uncertainties. This set of research works has produced several papers in international journals and peer-reviewed conferences.
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Smith, Amy. "Multi-scale modelling of blood flow in the coronary microcirculation." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:e6f576a2-75d9-4778-a640-a1e8551141a6.
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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.
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Kyrke-Smith, Teresa Marie. "Ice-stream dynamics : the coupled flow of ice sheets and subglacial meltwater." Thesis, University of Oxford, 2014. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.629515.
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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.
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Zhong, Yiming. "Modelling sediment transportation and overland flow." Thesis, University of Oxford, 2013. http://ora.ox.ac.uk/objects/uuid:a45eefae-5a0f-4917-9abb-261ae792f2ee.
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The erosion and transport of fertile topsoil is a serious problem in the U.S., Australia, China and throughout Europe. It results in extensive environmental damage, reduces soil fertility and productivity, and causes significant environmental loss. It is as big a threat to the future sustainability of global populations as climate change, but receives far less attention. With both chemicals (fertilizers, pesticides, herbicides) and biological pathogens (bacteria, viruses) preferentially sorbing to silt and clay sized soil particles, estimating contaminant fluxes in eroded soil also requires predicting the transported soils particle size distribution. The Hairsine-Rose (HR) erosion model is considered in this thesis as it is one of the very few that is specifically designed to incorporate the effect of particle size distribution, and differentiates between non-cohesive previously eroded soil compared with cohesive un-eroded soil. This thesis develops a new extended erosion model that couples the HR approach with the one-dimensional St Venant equations, and an Exner bed evolution equation to allow for feedback effects from changes in the local bed slope on surface hydraulics and erosion rates to be included. The resulting system of 2I +3 (where I = number of particle size classes) nonlinear hyperbolic partial differential equations is then solved numerically using a Liska-Wendroff predictor corrector finite difference scheme. Approximate analytical solutions and series expansions are derived to overcome singularities in the numerical solutions arising from either boundary or initial conditions corresponding to a zero flow depth. Three separate practical applications of the extended HR model are then considered in this thesis, (i) flow through vegetative buffer strips, (ii) modelling discharge hysteresis loops and (iii) the growth of antidunes, transportational cyclic steps and travelling wave solutions. It is shown by comparison against published experimental flume data that predictions from the extended model are able to closely match measurements of deposited sediment distribution both upstream and within the vegetative buffer strip. The experiments were conducted with supercritical inflow to the flume which due to the increased drag from the vegetative strip, resulted in a hydraulic jump just upstream of the vegetation. As suspended sediment deposited at the jump, this resulted in the jump slowly migrating upstream. The numerical solutions were also able to predict the position and hydraulic jump and the flow depth throughout the flume, including within the vegetative strip, very well. In the second application, it is found that the extended HR model is the first one that can produce all known types of measured hysteresis loops in sediment discharge outlet data. Five main loop types occur (a) clockwise, (b) counter-clockwise, (c,d) figure 8 of both flow orientations and (e) single curve. It is clearly shown that complicated temporal rainfall patterns or bed geometry are not required to developed complicated hysteresis loops, but it is the spatial distribution of previously eroded sediment that remains for the start of a new erosion event, which primarily governs the form of the hysteresis loop. The role of the evolution of the sediment distribution in the deposited layer therefore controls loop shape and behavior. Erosion models that are based solely on suspended sediment are therefore unable to reproduce these hysteretic loops without a priori imposing a hysteretic relationship on the parameterisations of the erosion source terms. The rather surprising result that the loop shape is also dominated by the suspended concentration of the smallest particle size is shown and discussed. In the third application, a linear stability analysis shows that instabilities, antidunes, will grow and propagate upstream under supercritical flow conditions. Numerical simulations are carried out that confirm the stability analysis and show the development and movement of antidunes. For various initial parameter configurations a series of travelling antidunes, or transportational cyclic steps, separated by hydraulic jumps are shown to develop and evolve to a steady form and wave speed. Two different forms arise whereby (a) the deposited layer completely shields the underlying original cohesive soil so that the cohesive layer plays no role in the speed or shape of the wave profile or (b) the cohesive soil is exposed along the back of the wave such that both the non-cohesive and cohesive layers affect the wave profile. Under (a) the solutions are obtained up to an additive constant as the actual location of the boundary of the cohesive soil is not required, whereas for (b) this constant must be determined in order to find the location on the antidune from where the cohesive soil becomes accessible. For single size class soils the leading order travelling wave equations are fairly straightforward to obtain for both cases (a) and (b). However for multi-size class soils, this becomes much more demanding as up to 2I + 3 parameters must be found iteratively to define the solution as each size class has its own wave profile in suspension and in the antidune.
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McCormick, Matthew. "Ventricular function under LVAD support." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:0d49ba30-b508-4c69-9ba6-b398d4338c01.
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This thesis presents a finite element methodology for simulating fluid–solid interactions in the left ventricle (LV) under LVAD support. The developed model was utilised to study the passive and active characteristics of ventricular function in anatomically accurate LV geometries constructed from normal and patient image data. A non–conforming ALE Navier–Stokes/finite–elasticity fluid–solid coupling system formed the core of the numerical scheme, onto which several novel numerical additions were made. These included a fictitious domain (FD) Lagrange multiplier method to capture the interactions between immersed rigid bodies and encasing elastic solids (required for the LVAD cannula), as well as modifications to the Newton–Raphson/line search algorithm (which provided a 2 to 10 fold reduction in simulation time). Additional developments involved methods for extending the model to ventricular simulations. This required the creation of coupling methods, for both fluid and solid problems, to enable the integration of a lumped parameter representation of the systemic and pulmonary circulatory networks; the implementation and tuning of models of passive and active myocardial behaviour; as well as the testing of appropriate element types for coupling non–conforming fluid– solid finite element models under high interface tractions (finding that curvilinear spatial interpolations of the fluid geometry perform best). The behaviour of the resulting numerical scheme was investigated in a series of canonical test problems and found to be convergent and stable. The FD convergence studies also found that discontinuous pressure elements were better at capturing pressure gradients across FD boundaries. The ventricular simulations focused firstly on studying the passive diastolic behaviour of the LV both with and without LVAD support. Substantially different vortical flow features were observed when LVAD outflow was included. Additionally, a study of LVAD cannula lengths, using a particle tracking algorithm to determine recirculation rates of blood within the LV, found that shorter cannulas improved the recirculation of blood from the LV apex. Incorporating myocardial contraction, the model was extended to simulate the full cardiac cycle, converging on a repeating pressure–volume loop over 2 heart beats. Studies on the normal LV geometry found that LVAD implementation restricts the recirculation of early diastolic inflow, and that fluid–solid coupled models introduce greater heterogeneity of myocardial work than was observed in equivalent solid only models. A patient study was undertaken using a myocardial geometry constructed using image data from an LVAD implant recipient. A series of different LVAD flow regimes were tested. It was found that the opening of the aortic valve had a homogenising effect on the spatial variation of work, indicating that the synchronisation of LVAD outflow with the cardiac cycle is more important if the valve remains shut. Additionally, increasing LVAD outflow during systole and decreasing it during diastole led to improved mixing of blood in the ventricular cavity – compared with either the inverse, or holding outflow constant. Validation of these findings has the potential to impact the treatment protocols of LVAD patients.
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Frenander, Hannes. "High-order finite difference approximations for hyperbolic problems : multiple penalties and non-reflecting boundary conditions." Doctoral thesis, Linköpings universitet, Beräkningsmatematik, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-134127.
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In this thesis, we use finite difference operators with the Summation-By-Partsproperty (SBP) and a weak boundary treatment, known as SimultaneousApproximation Terms (SAT), to construct high-order accurate numerical schemes.The SBP property and the SAT’s makes the schemes provably stable. The numerical procedure is general, and can be applied to most problems, but we focus on hyperbolic problems such as the shallow water, Euler and wave equations. For a well-posed problem and a stable numerical scheme, data must be available at the boundaries of the domain. However, there are many scenarios where additional information is available inside the computational domain. In termsof well-posedness and stability, the additional information is redundant, but it can still be used to improve the performance of the numerical scheme. As a first contribution, we introduce a procedure for implementing additional data using SAT’s; we call the procedure the Multiple Penalty Technique (MPT). A stable and accurate scheme augmented with the MPT remains stable and accurate. Moreover, the MPT introduces free parameters that can be used to increase the accuracy, construct absorbing boundary layers, increase the rate of convergence and control the error growth in time. To model infinite physical domains, one need transparent artificial boundary conditions, often referred to as Non-Reflecting Boundary Conditions (NRBC). In general, constructing and implementing such boundary conditions is a difficult task that often requires various approximations of the frequency and range of incident angles of the incoming waves. In the second contribution of this thesis,we show how to construct NRBC’s by using SBP operators in time. In the final contribution of this thesis, we investigate long time error bounds for the wave equation on second order form. Upper bounds for the spatial and temporal derivatives of the error can be obtained, but not for the actual error. The theoretical results indicate that the error grows linearly in time. However, the numerical experiments show that the error is in fact bounded, and consequently that the derived error bounds are probably suboptimal.
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Leonard, Katherine H. L. "Mathematical and computational modelling of tissue engineered bone in a hydrostatic bioreactor." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:05845740-1a74-4e19-95ea-6b5229d1af27.
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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.
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Figueroa, Leonardo E. "Deterministic simulation of multi-beaded models of dilute polymer solutions." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:4c3414ba-415a-4109-8e98-6c4fa24f9cdc.
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We study the convergence of a nonlinear approximation method introduced in the engineering literature for the numerical solution of a high-dimensional Fokker--Planck equation featuring in Navier--Stokes--Fokker--Planck systems that arise in kinetic models of dilute polymers. To do so, we build on the analysis carried out recently by Le~Bris, Leli\`evre and Maday (Const. Approx. 30: 621--651, 2009) in the case of Poisson's equation on a rectangular domain in $\mathbb{R}^2$, subject to a homogeneous Dirichlet boundary condition, where they exploited the connection of the approximation method with the greedy algorithms from nonlinear approximation theory explored, for example, by DeVore and Temlyakov (Adv. Comput. Math. 5:173--187, 1996). We extend the convergence analysis of the pure greedy and orthogonal greedy algorithms considered by Le~Bris, Leli\`evre and Maday to the technically more complicated situation of the elliptic Fokker--Planck equation, where the role of the Laplace operator is played out by a high-dimensional Ornstein--Uhlenbeck operator with unbounded drift, of the kind that appears in Fokker--Planck equations that arise in bead-spring chain type kinetic polymer models with finitely extensible nonlinear elastic potentials, posed on a high-dimensional Cartesian product configuration space $\mathsf{D} = D_1 \times \dotsm \times D_N$ contained in $\mathbb{R}^{N d}$, where each set $D_i$, $i=1, \dotsc, N$, is a bounded open ball in $\mathbb{R}^d$, $d = 2, 3$. We exploit detailed information on the spectral properties and elliptic regularity of the Ornstein--Uhlenbeck operator to give conditions on the true solution of the Fokker--Planck equation which guarantee certain rates of convergence of the greedy algorithms. We extend the analysis to discretized versions of the greedy algorithms.
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Hennessy, Matthew Gregory. "Mathematical problems relating to the fabrication of organic photovoltaic devices." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:7753abec-bb6e-4d8a-aa5b-b527c5beb49b.
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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.
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Barker, Tobias. "Uniqueness results for viscous incompressible fluids." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:db1b3bb9-a764-406d-a186-5482827d64e8.
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First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calderón. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L<sub>∞</sub>(-1; 0; L<sup>3, β</sup>(B(1) ⋂ ℝ<sup>3</sup> <sub>+</sub>)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new ϵ-regularity criterion. Third, we show that if a weak Leray-Hopf solution in ℝ<sup>3</sup> <sub>+</sub>×]0,∞[ has a finite blow-up time T, then necessarily lim<sub>t↑T</sub>||v(·, t)||<sub>L<sup>3,β</sup>(ℝ<sup>3</sup> <sub>+</sub>)</sub> = ∞ with 3 < β < ∞. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and Šverák [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in ℝ<sup>3</sup>, with solenoidal initial data in the critical Besov space ?<sup>-1/4</sup><sub>4,∞</sub>(ℝ<sup>3</sup>), which has certain continuity properties with respect to weak* convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in ℝ<sup>3</sup>×]0,∞[ has a finite blow-up time T, then necessarily lim<sub>t↑T</sub> ||v(·, t)||<sub>L<sub>3</sub>(ℝ<sup>3</sup>)</sub> = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.
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Pitters, Hermann-Helmut. "Aspects of exchangeable coalescent processes." Thesis, University of Oxford, 2015. http://ora.ox.ac.uk/objects/uuid:dbd83051-cffa-4fc9-b33f-59f837d8a9c2.
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In mathematical population genetics a multiple merger <i>n</i>-coalescent process, or <i>Λ</i> <i>n</i>-coalescent process, {<i>Π<sup>n</sup>(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<sub>n</sub></i>, respectively the total branch length, <i>L<sub>n</sub></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<sub>j</sub></i> count the number of mutations that are shared by <i>j</i> individuals in the sample. The random vector (<i>M<sub>1</sub></i>,...,<i>M<sub>n-1</sub></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>#Π<sup>n</sup>(t), t</i> ≥ 0} recording the number <i>#Π<sup>n</sup>(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.
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van, den Bremer T. S. "The induced mean flow of surface, internal and interfacial gravity wave groups." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:e735afe7-a77d-455d-a560-e869a9941f69.
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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.
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Yang, Xin-She. "Mathematical modelling of compaction and diagenesis in sedimentary basins." Thesis, University of Oxford, 1997. http://ora.ox.ac.uk/objects/uuid:0bdc6c43-4534-4f08-97e2-8a33d6b13e61.
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Sedimentary basins form when water-borne sediments in shallow seas are deposited over periods of millions of years. Sediments compact under their own weight, causing the expulsion of pore water. If this expulsion is sufficiently slow, overpressuring can result, a phenomenon which is of concern in oil drilling operations. The competition between pore water expulsion and burial is complicated by a variety of factors, which include diagenesis (clay dewatering), and different modes (elastic or viscous) of rheological deformation via compaction and pressure solution, which may also include hysteresis in the constitutive behaviours. This thesis is concerned with models which can describe the evolution of porosity and pore pressure in sedimentary basins. We begin by analysing the simplest case of poroelastic compaction which in a 1-D case results in a nonlinear diffusion equation, controlled principally by a dimensionless parameter lambda, which is the ratio of the hydraulic conductivity to the sedimentation rate. We provide analytic and numerical results for both large and small lambda in Chapter 3 and Chapter 4. We then put a more realistic rheological relation with hysteresis into the model and investigate its effects during loading and unloading in Chapter 5. A discontinuous porosity profile may occur if the unloaded system is reloaded. We pursue the model further by considering diagenesis as a dehydration model in Chapter 6, then we extend it to a more realistic dissolution-precipitation reaction-transport model in Chapter 7 by including most of the known physics and chemistry derived from experimental studies. We eventually derive a viscous compaction model for pressure solution in sedimentary basins in Chapter 8, and show how the model suggests radically different behaviours in the distinct limits of slow and fast compaction. When lambda << 1, compaction is limited to a basal boundary layer. When lambda >> 1, compaction occurs throughout the basin, and the basic equilibrium solution near the surface is a near parabolic profile of porosity. But it is only valid to a finite depth where the permeability has decreased sufficiently, and a transition occurs, marking a switch from a normally pressured environment to one with high pore pressures.
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Stevens, Ben. "Short-time structural stability of compressible vortex sheets with surface tension." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:378713da-cd05-4b9a-856d-bee2b0fb47ce.
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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.