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The flow behaviour of liquid metals (Sn, Pb and Sn-Pb eutectic) under different shearing conditions is investigated. Experiments were performed with two designs of concentric cylinder viscometers: rotating the inner cylinder (Searle) and rotating the outer cylinder (Couette). The latter technique is uncommon and the equipment was optimised with standard oils. The flow behaviour for the metals differs in the two systems. The curves of 'apparent' viscosity versus shear rate may be divided into two regimes: I. At lower shear rates (<200 s-1): a reduction of 'apparent' viscosity with shear was observed with both viscometers. It is suggested that the high density and high surface tension of the metals and eccentricity between the cylinders at low shear rates, leads to instabilities. Results at low shear rates were therefore discarded and further detailed analysis would be required for a fuller understanding of this behaviour. II. At higher shear rates: a steady, shear-independent behaviour of 'apparent' viscosity with shear rate is observed in the Couette system (upto 600 s-1) whereas in the Searle system the 'apparent' viscosity increases with shear rate (upto 2600 s-1). From hydrodynamic theory about Newtonian fluids, it is suggested that in the Searle type viscometer, the fluid is unstable and Taylor vortices are expected at low shear rates (~80 s-1). This gives rise to an increase in the 'apparent' viscosity with shear rate. Whereas, in the Couette type, the flow is more stable, resulting in a steady 'apparent' viscosity. This interpretation is consistent with liquid metals behaving as Newtonian fluids, but further research is required to confirm this. The author suggests further experiments, with the prime one being the investigation of the fluid with counter and co-rotation of the cylinders in order to observe more complex flows. The results are expected to have implications in the modelling of flow for liquid metal processes, especially the initiation of Taylor vortices under the unstable flow conditions produced by rotating the inner cylinder.

We are interested in the onset of instability of the axisymmetric flow between two concentric spherical shells that differentially rotate about a common axis in the narrow-gap limit. The expected mode of instability takes the form of roughly square axisymmetric Taylor vortices which arise in the vicinity of the equator and are modulated on a latitudinal length scale large compared to the gap width but small compared to the shell radii. At the heart of the difficulties faced is the presence of phase mixing in the system, characterised by a non-zero frequency gradient at the equator and the tendency for vortices located off the equator to oscillate. This mechanism serves to enhance viscous dissipation in the fluid with the effect that the amplitude of any initial disturbance generated at onset is ultimately driven to zero. In this thesis we study a complex Ginzburg-Landau equation derived from the weakly nonlinear analysis of Harris, Bassom and Soward [D. Harris, A. P. Bassom, A. M. Soward, Global bifurcation to travelling waves with application to narrow gap spherical Couette flow, Physica D 177 (2003) p. 122-174] (referred to as HBS) to govern the amplitude modulation of Taylor vortex disturbances in the vicinity of the equator. This equation was developed in a regime that requires the angular velocities of the bounding spheres to be very close. When the spherical shells do not co-rotate, it has the remarkable property that the linearised form of the equation has no non-trivial neutral modes. Furthermore no steady solutions to the nonlinear equation have been found. Despite these challenges Bassom and Soward [A. P. Bassom, A. M. Soward, On finite amplitude subcritical instability in narrow-gap spherical Couette flow, J. Fluid Mech. 499 (2004) p. 277-314] (referred to as BS) identified solutions to the equation in the form of pulse-trains. These pulse-trains consist of oscillatory finite amplitude solutions expressed in terms of a single complex amplitude localised as a pulse about the origin. Each pulse oscillates at a frequency proportional to its distance from the equatorial plane and the whole pulse-train is modulated under an envelope and drifts away from the equator at a relatively slow speed. The survival of the pulse-train depends upon the nonlinear mutual-interaction of close neighbours; as the absence of steady solutions suggests, self-interaction is inadequate. Though we report new solutions to the HBS co-rotation model the primary focus in this work is the physically more interesting case when the shell velocities are far from close. More specifically we concentrate on the investigation of BS-style pulse-train solutions and, in the first part of this thesis, develop a generic framework for the identification and classification of pulse-train solutions. Motivated by relaxation oscillations identified by Cole [S. J. Cole, Nonlinear rapidly rotating spherical convection, Ph.D. thesis, University of Exeter (2004)] whilst studying the related problem of thermal convection in a rapidly rotating self-gravitating sphere, we extend the HBS equation in the second part of this work. A model system is developed which captures many of the essential features exhibited by Cole's, much more complicated, system of equations. We successfully reproduce relaxation oscillations in this extended HBS model and document the solution as it undergoes a series of interesting bifurcations.

The results of an experimental study of phenomena which occur in the flow of a viscous fluid in closed domains with discrete symmetries are presented. The purpose is to investigate the role which ideas from low-dimensional dynamical systems have to play in describing qualitative changes that take place with variation of the governing parameters. Such a descriptive framework already exists for the case of the Taylor-Couette system, where the domain possesses a continuous azimuthal symmetry group. The present investigation is aimed at establishing the typicality of previously reported behaviour under progressive reductions of azimuthal symmetry. In the first investigations, the fixed outer circular cylinder of the standard system is replaced with one of square cross-section. Thus there is now discrete Ζ4 symmetry in the azimuthal direction. Knowledge of the two-dimensional flow field is used to establish the nature of the steady three-dimensional motion equivalent to Taylor vortex flow. It is shown that similar bifurcation sequences exist in both standard and square systems for the case of very small aspect ratio where a single Taylor cell is formed. This flow develops as the result of a bifurcation which breaks the Ζ2 symmetry that is imposed on the annulus by two solid stationary ends. The study is then extended to consider time-dependent effects in the square system. Two different oscillatory single-cell flows are identified, and it is shown that each is the result of a Hopf bifurcation. Selection of a particular dynamic mode is found to depend on the aspect ratio of the system. A low-dimensional bifurcation structure is uncovered which connects the two modes in parameter space, and involves a novel type of steady single-cell flow. Finally, observations are reported of a nontrivial type of dynamical behaviour which bears strong resemblance to motion found in a circularly symmetric Taylor-Couette system that is related to the Šilnikov mechanism for finite-dimensional chaos. A second variant on the Taylor-Couette system is considered where the outer cylinder is shaped like a stadium. The effect is to reduce further the overall symmetry of the domain to a Ζ2 × Ζ2 group. The two-dimensional flow field is investigated using both numerical and experimental techniques. Time-dependent phenomena are then investigated in the three-dimensional flow over a relatively wide range of aspect ratio. It is found that a sequence of a Hopf bifurcation followed by period-doubling bifurcations exists up to a certain aspect ratio, beyond which there is an apparently sudden and reversible transition between regular and irregular dynamical behaviour. Although this transition is not of a low-dimensional nature, the experimental results suggest that it exists as the result of a coalescence of the bifurcations which are found at lower values of aspect ratio.

As the world population approaches seven billion, a greater strain is put on the resources necessary to sustain life. One of the most basic and essential resources is water and while two thirds of the earth is covered by water, the majority is either salt water (oceans and seas) or it is too contaminated to drink. The purpose of this project is to develop a portable device capable of testing whether a specific source of water (i.e. lake, river, well…) is potable. There are numerous filtration techniques that can remove contaminants and make even the dirtiest water clean enough for consumption but they are for the most part, very time consuming and immobile processes. The device is not a means of water purification but rather focuses on determining the content of the water and whether it is safe. Particles within the water are separated and trapped using a combination of a Taylor Couette fluid flow system and Dielectrophoretic electrodes. This paper explores Taylor Couette flow in a large gap and low aspect ratio system through theory and experimentation with early stage prototypes. Different inner cylinder radii, 2.12cm, 1.665cm and 1.075cm, were tested at different speeds approaching, at and passing the critical Taylor number, 3825, 4713 and 6923 respectively for each cylinder. Dielectrophoretic (DEP) electrodes were designed, fabricated, coated and tested using latex beads to determine the method of integrating them within the fluid flow system. Taylor Couette theory, in terms of the formation of vortices within the large gap, small aspect ratio system, was not validated during testing. The flow pattern generated was more akin to a chaotic circular Couette flow but still served to move the particles toward the outer wall. Fully integrated tests were run with limited success. Recommendations were made to pursue both circular Couette flow as the basis for iv particle separation and dimensional changes in the setup to allow for the formation of Taylor vortices by increasing the radius ratio but still allowing for a larger volume of fluid.