Impeller-stirred mixing is one of the most important processes employed throughout the chemical, metallurgical and allied industries. The research reported in this thesis is focused on impeller stirred mixing associated with the refining of lead bullion. The aim of this process is to sequentially remove contained impurities such as copper, antimony, silver and bismuth. This occurs in hemispherical vessels, called kettles, where reagents are initially added to the lead bath to form surface dross that contains both the required impurity and a large amount of lead oxide. This dross is then continuously mixed back into the bath to remove the lead oxide and capture more of the required impurity. A key requirement for this process is to obtain and remove dross that contains a high concentration of the impurity. Although this process has been in operation for many years, there is very little known on how the fluid dynamics associated with the mixing process affects final dross content. The aim of this research is to fully investigate the lead refining process using scientific analysis methods that help understand the mixing process and provide design tools which can be used to optimise process conditions. The three methods of analysis used are: (1) Direct readings from a real kettle, (2) Physical modelling (using water), and finally (3) Computational Fluid Dynamics (CFD). The use of physical modelling, exploiting the techniques of similitude, to predict vortex was also validated. An Acoustic Doppler Velocimeter (ADV) probe was used for the velocity measurement at various locations inside the water model and this gave valuable insight about the flow phenomena occurring inside the refining kettle. A particular important finding was that when fluid is stirred above certain rotational speed the vortex depth becomes independent of the Reynolds number of the operation. With regards CFD technology, the Volume of Fluid (VOF) method was used to capture the free surface and the Lagrangian Particle Tracking (LPT) and Algebraic Slip Model (ASM) to simulate the dross phase. Appropriate methods were also used to represent the moving impeller region. Validation of simulation results against experimental data was very encouraging. Computed vortex depth showed the similar trend as observed during the experiments on the physical model. A design strategy was developed that integrates results from both physical and computational modelling to allow optimal process conditions to be predicted at the kettle design stage. The use of this integrated physical and computational modelling methodology successfully helped eliminate surface swirl by introducing baffles to the kettle. The design and introduction of these flow controllers was also validated to ensure that it optimised the dross mixing process and final impurity content in the dross.

In this work we use the Zakharov-Kuznetsov equation to study the evolution of a plane soliton subjected to a two-dimensional perturbation. The first part of the thesis is concerned with determining the growth rate of such a perturbation. We present two closely related methods which allow us to obtain rigorously the growth rates much more directly and simply than previous approaches. Both methods are general and hence applicable to other problems. If the perturbation is of a large enough wavelength, the plane soliton will evolve into more stable coherent structures of the form of two-dimensional solitons. This process is the subject of the remainder of the thesis. A weakly nonlinear analysis which fully describes the preliminary stages of the process is developed. We have studied how the eventual fate of a plane soliton is affected by the wavelength of the perturbation and obtained a simple formula for the variation of the number of cylindrical solitons formed with this wavelength. The methods developed in this thesis have been used to obtain an analytical description of a soliton state that occurs in coupled optical fibres.

Bosonic string theory and superstring theory are briefly overviewed. Three dimensional field theories are similarly discussed, with a focus on effective $N=2$ supersymmetric theories. It is shown how to induced contributions to the Chern-Simons level of the low energy theory, by integrating out massive matter. Such effective field theories are then shown to arise from type IIB brane configurations based on the Hanany-Witten brane configuration. Strong-weak dualities are overviewed, leading to a discussion of the three dimensional strong-weak dualities: Aharony duality for theories with zero Chern-Simons level, and Giveon-Kutasov duality for theories with non-zero Chern-Simons level. In the results section, brane configurations corresponding to three-dimensional $N=2$ $U(N_{c})$ field theories with various numbers of flavour of massive matter are investigated. The resulting low energy field theories are explained, and the flows between Aharony and Giveon-Kutasov dualities are catalogued. Three dimensional $N=2$ effective field theories obtained through the inclusion of massive adjoint matter are also examined, with the flows between Aharony and Giveon-Kutasov dualities, again, catalogued. Finally, the significance of the results and the possibilities for future research, are discussed.

This study provides an understanding of various aspects of hydrogeological systems modelling and the use of computational techniques to predict and optimise hydrological parameter assessment, anisotropic scaling, macrodispersion and solute flux measurements under unsteady, uniform/non-uniform flow conditions. The incorporated models are structured around multi-physics continuum mechanics analysis to investigate fluid flow and solute transport in hydrogeological systems. The control-volume unstructured mesh configuration, based on cell-centred or vertex-based FV algorithms for CFD and CSM problems is employed. The non-linear material behaviour exhibited by porous soils and the fluid flow evaluation under system stresses is described by elasto-visco-plastic constitutive relationships and the coupling between CFD and CSM processes. The designed simulation models are used to calibrate the flow problems associated with regional groundwater levels estimation, determination of soil hydraulic properties and moisture distribution in dry soils in response to infiltration of compressible or incompressible fluids. For solute transport problems, investigations of spatial distribution of solute species in homogeneous/layered heterogeneous systems are undertaken by accounting for chemical, geochemical and biological reactions caused by particle deposition processes and liquid-solid interactions in natural subsurface systems. The simulated shape and spread of contaminant plume are effectively influenced by the governing transport mechanism for solutes. The attention in leachate is predicted to have a significant role in reducing the level of contaminant concentration and its potential impact on the attainable groundwater resources.

This thesis describes the development of a methodology to determine large-scale and meso-scale atmospheric dispersion patterns. The research is only concerned with outdoor exposure to atmospheric pollutants and aims to identify pollution sources using dispersion modelling with the assistance of ground level measurements from British, French and other monitoring stations and remote sensing technology. Lagrangian Particle Dispersion (LPD) models compute trajectories of a large number of notional particles and can be used to numerically simulate the dispersion of a pollutant (passive tracer) in the planetary boundary layer. Two widely used atmospheric dispersion models were employed: the Hybrid Single Particle Lagrangian Integrated Trajectory (HYSPLIT) model by R. Draxler, and the model FLEXPART by Stohl et al. Both models possess forward tracking and inverse (or receptor-based) modes. Meteorological data output from the PSU/NCAR Mesoscale model (known as MM5), or datasets from the European Centre of Medium-range Weather Forecast (ECMWF) are used to drive the dispersion models. Linkage routines were developed to interpret the LPD codes with the required meteorological information. This study aims to determine whether current approaches and practice for atmospheric dispersion modelling are reliable, consistent and up-to-date. An intercomparison of the models FLEXPART and HYSPLIT is performed for known episodes to determine their accuracy, ease of use, effect of source specification and to investigate their sensitivity to input data and mesh resolution, and in particular the effect of different model formulations and assumptions followed by the models. The possibility of identifying emission sources in the near and far field is investigated, by modelling dispersion backwards in time, in particular the discrimination of multiple sources from receptor data is discussed. The effect of meteorological data resolution on the output of LPD models was evaluated and the most suitable methodology for better source definition was determined for different modelling scales, ranging from the intercontinental transport of airborne pollutants to simulating pollution episodes caused by local sources.

We consider a model for random loops on graphs which is inspired by the Feynman–Kac formula for the grand canonical partition function of an ideal gas. We associate to this model a corresponding occupation field, which is a positive random field detailing the total time spent by loops at each vertex. We argue that well known critical phenomena for the ideal gas can be reinterpreted in terms of random variables of this occupation field. We also argue that higher order correlations, such as the existence of off-diagonal long-range order, can only be seen in the occupation field by studying a modified space–time model of loops. We provide an isomorphism theorem for this model to a complex Gaussian field, and derive a version of Symanzik’s formula which describes the ideal gas interacting with a random background environment. Finally we consider the effect of interactions on the gas, and present a large deviations analysis of the cycle distribution of the loop model under two mean field Hamiltonians.

This thesis is concerned with the classification and evaluation of various numerical schemes that are available for computing solutions for fluid-flow problems, and secondly, with the development of an improved numerical discretisation scheme of the finite-volume type for solving steady-state differential equations for recirculating flows with and without sources. In an effort to evaluate the performance of the various numerical schemes available, some standard test cases were used. The relative merits of the schemes were assessed by means of one-dimensional laminar flows and two-dimensional laminar and turbulent flows, with and without sources. Furthermore, Taylor series expansion analysis was also utilised to examine the limitations that were present. The outcome of this first part of the work was a set of conclusions, concerning the accuracy of the numerous schemes tests, vis-a-vis their stability, ease of implementation, and computational costs. It is hoped that these conclusions can be used by `computational fluid-dynamics' practitioners in deciding on an optimum choice of scheme for their particular problem. From the understanding gained during the first part of the study, and in an effort to combine the attributes of a successful discretisation scheme, eg positive coefficients. conservation and the elimination of 'false-diffusion', a new flow-oriented finite-volume numerical scheme was devised and applied to several test cases in order to evaluate its performance. The novel approach in formulating the new CUPID* scheme (for Corner UPw^nDing) underlines the idea of focussing attention at the control-volume corners rather than at the control-volume cell-faces. In two-dimensions, this leads to an eight neighbour influence for the central grid point value, depending on the flow-directions at the corners of the control-volume. In the formulation of the new scheme, false-diffusion is considered from a pragmatic perspective, with emphasis on physics rather than on strict mathematical considerations such as the order of discretisation, etc. The accuracy of the UPSTREAM scheme (for JJPwind in STREAMIines) indicates that although it is formally only first-order accurate, it considerably reduces 'false-diffusion'. Scalar transport calculations (without sources) show that the UPSTREAM scheme predicts bounded solutions which are more accurate than the upwind-difference scheme and the unbounded skew-upstream-difference scheme. Furthermore, for laminar and turbulent flow calculations, improved results are obtained when compared with the performances of the other schemes. The advantage of the UPSTREAM-difference scheme is that all the influence coefficients are always positive and thus the coefficient matrices are suitable for iterative solution procedures. Finally, the stability and convergence characteristics are similar to those of the upwind-difference scheme, eg converged solutions are guaranteed. What cannot be guaranteed, however, is the conservatism of the scheme and it is recommended that future work should be directed towards improving that disadvantage.

This thesis examines how periodic orbits may be used in acoustics to understand solutions of the Helmholtz equation. A review of the links between ray and wave mechanics is given including WKBJ (Wentzel, Kramers, Brillouin and Jeffreys) and EBK (Einstein, Brillouin, Keller) methods. It is also noted that some mode shapes in chaotic enclosures are scarred by the short periodic orbits. This motivates the proposal of the Mode Scar Hypothesis and the Mode Resonance Function Hypothesis. The trace formula, which is a sum over periodic orbits, approximates the level density for an acoustic enclosure. The trace formula in the concentric annulus domain is derived using a formulation for enclosures with continuous symmetry by Creagh and Littlejohn [1]. Results for the variance of the difference between the true and average mode counts are obtained. A technique called short periodic orbit theory (SPOT) for the approximation of mode shapes devised by Babiˇc and Buldyrev [2] and Vergini [3] is given. SPOT is extended to impedance boundary conditions. SPOT is implemented in the quarter stadium, quadrupole, circle and eccentric annulus enclosures with Dirichlet, Neumann and impedance boundary conditions. Concave enclosures with Dirichlet or Neumann boundary conditions were best approximated using SPOT. A design loop for enclosures is proposed using the periodic orbit ideas given. A model problem is used to provide insight into the effectiveness of these methods. It was found that it was not possible to breakdown all mode shapes in the eccentric annulus into contributions from short periodic orbits.

This thesis is devoted to the application of the Finite Difference Time Domain (FDTD) method for describing light interaction with liquid crystal media. This method has been known in the electromagnetic community since 1966, but has so far only found limited application in liquid crystal optics. The thesis consists of four parts. In the first part of the thesis I extend the conventional FDTD algorithm to be used for studying anisotropic media with continuous spatial variations of the dielectric properties. In the second part I discuss tests of the algorithm and software on various dielectric systems with known response. The third and the fourth parts of the thesis are concerned with the nonlinear interaction between light and liquid crystals. In the third part I develop a FDTD self-consistent algorithm that takes into account the coupling between light and liquid crystal orientation. The algorithm is used to simulate the optical Freedericksz transition in a homeotropic liquid crystal cell. I find that solving the problem self-consistently significantly modifies the main characteristics of the transition. In the fourth part I use the liquid crystal FDTD algorithm to investigate the birth of optical singularities. The results are compared to a recent analytical theory. I find that the analytic theory is only qualitatively useful except in the extreme short wave limit.

Solar sailing is the use of a thin and lightweight membrane to reflect sunlight and obtain a thrust force on the spacecraft. That is, a sailcraft has a potentially-infinite specific impulse and, therefore, it is an attractive solution to reach mission goals otherwise not achievable, or very expensive in terms of propellant consumption. The recent scientific interest in near-Earth asteroids (NEAs) and the classification of some of those as potentially hazardous asteroids (PHAs) for the Earth stimulated the interest in their exploration. Specifically, a multiple NEA rendezvous mission is attractive for solar-sail technology demonstration as well as for improving our knowledge about NEAs. A preliminary result in a recent study showed the possibility to rendezvous three NEAs in less than ten years. According to the NASA’s NEA database, more than 12,000 asteroids are orbiting around the Earth and more than 1,000 of them are classified as PHA. Therefore, the selection of the candidates for a multiple-rendezvous mission is firstly a combinatorial problem, with more than a trillion of possible combinations with permutations of only three objects. Moreover, for each sequence, an optimal control problem should be solved to find a feasible solar-sail trajectory. This is a mixed combinatorial/optimisation problem, notoriously complex to tackle all at once. Considering the technology constraints of the DLR/ESA Gossamer roadmap, this thesis focuses on developing a methodology for the preliminary design of a mission to visit a number of NEAs through solar sailing. This is divided into three sequential steps. First, two methods to obtain a fast and reliable trajectory model for solar sailing are studied. In particular, a shape-based approach is developed which is specific to solar-sail trajectories. As such, the shape of the trajectory that connects two points in space is designed and the control needed by the sailcraft to follow it is analytically retrieved. The second method exploits the homotopy and continuation theory to find solar-sail trajectories starting from classical low-thrust ones. Subsequently, an algorithm to search through the possible sequences of asteroids is developed. Because of the combinatorial characteristic of the problem and the tree nature of the search space, two criteria are used to reduce the computational effort needed: (a) a reduced database of asteroids is used which contains objects interesting for planetary defence and human spaceflight; and (b) a local pruning is carried out at each branch of the tree search to discard those target asteroids that are less likely to be reached by the sailcraft considered. To reduce further the computational effort needed in this step, the shape-based approach for solar sailing is used to generate preliminary trajectories within the tree search. Lastly, two algorithms are developed which numerically optimise the resulting trajectories with a refined model and ephemerides. These are designed to work with minimum input required by the user. The shape-based approach developed in the first stage is used as an initial-guess solution for the optimisation. This study provides a set of feasible mission scenarios for informing the stakeholders on future mission options. In fact, it is shown that a large number of five-NEA rendezvous missions are feasible in a ten-year launch window, if a solar sail is used. Moreover, this study shows that the mission-related technology readiness level for the available solar-sail technology is larger than it was previously thought and that such a mission can be performed with current or at least near-term solar sail technology. Numerical examples are presented which show the ability of a solar sail both to perform challenging multiple NEA rendezvous and to change the mission en-route.