Some studies of fluid mixing and transport
In this thesis we study four problems with potential biological and industrial applications which rely on fluid mixing and transport. The problem of simultaneous ultrafiltration, diffusion and osmosis across a membrane separating two fluids is studied, numerically and asymptotically, as a model for an artificial kidney dialyser. Couplings between the different transport mechanisms prove significant in determining overall transport rates. Our model appears to be the first to treat the three transport mechanisms in a spatially structured framework, and shows that previous, spatially averaged models can overestimate transport rates. Our results can be used to optimise dialyser geometry and to profile dialysis sessions. The remainder of this thesis concerns some fundamentals of fluid mixing and mixer design. Techniques for assessing the quality of fluid mixing are reviewed, and applied to a two-dimensional laminar chaotic flow. We find no outright optimum mixing method across the range of measures, suggesting that `sieving' a collection of mixing methods according to increasingly complicated mixing measures may fail to identify a global optimum. `Topological chaos' appears to allow good mixing stretch rate to be built-in to batch mixer design, avoiding the need to tune the mixer parameters, provided a correct flow topology is created. We show that the theoretical stretch rate predictions are achieved quite tightly, in practice in a significant fraction of the flow domain; we investigate the practicalities of topologically chaotic mixers. Finally, we discuss whether topological chaos may also apply to three-dimensional static mixer design, in a braided pipe mixer, in which pipe flow is mixed around carefully designed twisted inner pipes. We expect such a device to mix well if the inner pipes have appropriate topology. However, we demonstrate how three-dimensional flow features can undermine mixing performance.