Three types of gas dispersion in a three-phase fluidized bed have been distinguished. Beds of large, high inertia, particle lead to bubble break-up and a good gas dispersion; conversely small particles give large bubbles; while the third type is an intermediate one whose behaviour depends on the voidage. A model of the interaction between a single bubble and a single particle in a fluidized bed is described. The two-dimensional equations of motion for the particle are solved numerically and the results are presented in dimensionless form as the particle impact velocity on the bubble surface versus the fluidization velocity. These results can be used for prediction of the type of gas dispersion for any bed of specified voidage, made up of particles and a liquid of known properties with low or moderate gas flow rates. Experiments with beds of 2mm diem, glass beads fluidized to various expansions by a liquid paraffin were carried out using two modes of gas injection, continuous and pulsed. High speed cine films were taken of the bubble dispersion emerging from the surface of the bed and analysed frame by frame to give bubble sizes and size distribution when gas was injected continuously. Similar films were taken of pulsed bubbles at the injection zone, rising through the bed and emerging from the surface. Typical sequences of photographs for each set of conditions are presented. Some sequences are of particular interest in showing the mechanism involved in bubble break-up. The experimental observations indicate that when this system is highly fluidized it behaves as a bubble break-up type of bed: conversely at low voidage there is little break-up and a greater tendency for bubble coalescence. The predictions from the theoretical model are shown to be in agreement with these experimental findings.

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The work reported in this Thesis involves studies of the extensional deformation of complex fluids. Complex fluids are so-called in terms of their composition, microstructure and rheology. Of particular interest in the work reported herein is the extensional viscosity of mobile (low shear viscosity) fluids in uniaxial extensional (or elongational flows) as this form of fluid determination is of wide concern within the process industries and is an important part of efforts in rheometry (the science of rheological measurement) to develop appropriate techniques for the measurement of extensional flow properties (in terms of a parameterisation of the resistance to extension). The principal experimental technique reported herein is a form of filament-stretching rheometer which generates a steady uniaxial extension of a fluid at the mid-point of a filament which is generated by the simultaneous motion of two confirming plate geometries. Difficulties which arise in this experiment and new approaches to their solution are described herein and the Thesis contains a comprehensive account of the design, construction, testing and refinement of such an instrument. Due to the relatively low rates of extension associated with filament-stretching rheometers, a new form of filament generation is described which involves fluid cavitation. Using this technique, which involves the generation of a rapidly stretching filament by the collapse of a gas-filled bubble, industrially relevant deformation rates are achieved and the technique has been successfully employed, for the first time, to provide a measure of resistance to extension of multigrade motor lubricants under process relevant conditions of fluid extension.

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This thesis presents procedures of the multigrid method using unstructured grids for solutions of inviscid compressible flows. The work concentrates on the components of the procedures, such as flow solvers (smoothers), intergrid interpolation, multigrid cycles, nested iterations and data structures. The performance and comparison of different smoothers are in particular emphasised. Some related aspects, e.g. the boundary treatment, are also discussed. Several two dimensional examples, for problems of external and internal inviscid compressible flows, are given to demonstrate the ability of the methods in improving the efficiency of simulating steady compressible flows.

A three dimensional mathematical model of a flexible fibre in a sheared suspension is proposed. The model, which includes fibre stiffness and distributed couples, was made up of differential force and moment balances from standard theory on three-dimensional bending curves. The viscous forces are calculated from the first term of the slender body theory. An inextensibility constraint is included to ensure the preservation of the arc length in time (19). The system is assumed to be quasi-static (inertialess) which makes it possible to separate the problem into a spatial boundary value at a time initial value problem. In the differential equations describing the fibre at each time, the derivatives are replaced by their equivalent finite difference approximations, giving a set of simultaneous algebraic equations which is solved by an LU decomposition band solver scheme. The solution of the boundary value problem yields the slip velocity at each node on the fibre, from which a new fibre position can be found by the time integration. The time integration is carried out using the modified Euler predictor-corrector method including a second order predictor formula. Automatic time-step size control is applied. This scheme was compared with Gear's method. The performance of the modified Euler method proved to be superior in this problem. The reliability of the model in physical terms was determined by comparing simulated fibre rotations in Couette flow with the equivalent real fibre rotations recorded in the laboratory. For the two fibre aspect ratios studied, the agreement was very good. In general when an experimental fibre configuration is not available, uncertainty exists about the initial condition to be used: In Couette flow, the obvious one of a straight fibre on the streamlines cannot be used; although the motion is there very slow, a small time-step is apparently still necessary to maintain stability. This makes it impractical to simulate fibre motion very close to the streamlines. The linear theory proposed by Hinch (19) for deformation decay rates of perfectly flexible, nearly straight fibres was reconfirmed, for Couette flow. The range of validity of that theory as a function of fibre stiffness was investigated. Results are presented on radial migration of a fibre suspended in Poiseuille flow. A qualitative description of the phenomenon and approximate quantitative results are obtained, and compared with experimental observations in the literature.

The basic difficulties associated with the numerical solution of the incompressible Navier-Stokes equations in primitive variables are identified and analysed. These difficulties, namely the lack of self-adjointness of the flow equations and the requirement of choosing compatible interpolations for velocity and pressure, are addressed with the development of consistent Petrov-Galerkin formulations. In particular, the solution of incompressible viscous flow problems using simple equal order interpolation for all variables becomes possible.

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