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This thesis considers the asymptotic and numerical solutions of two systems of partial differential equations which arise in fluid dynamics. Suitable numerical schemes are used for a number of different situations and the results are compared with asymptotic solutions when available.

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This thesis considers the use of NMR (Nuclear Magnetic Resonance) in investigating both the formulation, using microchannels, and characterisation of complex fluids using NMR techniques. Multiphase microchannel flows have many applications and understanding the hydrodynamics and mass transfer is essential for developing such systems. Time-averaged velocity images of the parallel flow of PDMS (polydimethylsiloxane) and water in a 500 μm x 250 μm microchannel were compared with an analytical solution and simulations. This was extended into imaging multiphase flows in a smaller (250 μm x 100 μm) microchannel. Poor signal-to-noise levels in this geometry led to the adoption of a spatially- and chemical-shift-resolved propagator acquisition. This was used to differentiate two different flow regimes: parallel flow and droplet production. RheoNMR is the use of NMR techniques in the study of rheology. Time-averaged velocity imaging was used to image a shear-banding micellar solution, cetylpyridinum chloride with sodium salicylate (CPyC1 / NaSal) in sodium chloride brine, in a variety of standard rheological geometries. The fast velocity measurement sequence GERVAIS was used to measure starting and stopping of flows of the CPyC1 / NaSal solution in a wide-gap Couette cell; these were compared to similar experiments performed using PDMS. The propagation of an elastic wave across the cell was seen during start-up and an unusual rebounding oscillation was evident in the stopped flow experiments in the CPyC1 sample. The fast diffusion measurement sequence, Difftrain, was modified to reduce the achievable incremental observation time (at the expense of slightly reduced chemical resolution). This was applied to two case studies: measuring the surface-to-volume ratio of a model porous medium and the droplet sizing of oil-in-water emulsions manufactured with fast-diffusing oils. Finally, NMR relaxometry and diffusometry were used to investigate a phase transition in the structure of a sample of a commercial face cream.

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