Thesis (PhD)--University of South Australia, 2000

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The fundamental theorem of projective geometry states that a mapping from a projective space to itself whose range has a sufficient number of points in general position is a projective transformation possibly combined with a self-homomorphism of the underlying field. We obtain generalisations of this in many directions, dealing with the case where the mapping is only defined on an open subset of the underlying space, or a subset of positive measure, and dealing with many different spaces over many different rings.

The aim of this work was to use simulation as a tool to better understand areas of orthokinetic (shear-induced) aggregation which are still not well understood. These areas include aggregate structure, aggregate strength, breakup and restructuring and combined perikinetic/orthokinetic aggregation. Previous simulation studies were reviewed and it was concluded that the methodology of Chen and Doi (1989) was an appropriate starting point for this study. The modified simulation was validated by comparison with theoretical and experimental results. Orthokinetic aggregates were found to have a fractal structure with an estimated value of 1.65. Scaling exponents, which were shown to be indicative of fractal dimension, of 2.1-2.7 were also obtained. Flexible bonds allowed restructuring to occur which led to an increase in the co-ordination number, scaling exponent, aggregate strength and a reduction in aggregate size. Thus aggregate strength increases with fractal dimension. It was confirmed that both restructuring and breakup/reformation could lead to the formation of small, compact aggregates. The high shear conditions simulated favoured breakup/reformation, while restructuring was expected to dominate with more flexible bonds, possibly at lower shear rates. Taking some account of hydrodynamic interactions by the inclusion of Kirkwood-Riseman theory led to an increase in the compactness of the aggregates and the co-ordination numbers, as well as a decrease in size of the aggregates. The results showed that hydrodynamic interactions can not be ignored. The explanation for the dramatic effects was that particles/microflocs on the outer edges of the aggregates broke off and reformed in a more compact way. Erosion was found to dominate in all cases, thus supporting the theory that erosion dominates at higher fractal dimensions. The shearing range simulated was found to be relatively high (equivalent to &lt200s-1 for particles of 2-5??m), producing relatively small aggregates. Hence it is proposed that under high shear conditions, erosion dominates. It was shown by extension of the DLCA algorithmic restructuring work of Meakin and Jullien (1988, 1989) that the scattering patterns observed in gently sheared aggregating systems are consistent with the interpretation that the shearing causes partial restructuring at large length scales.

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