Reduced models in fluid dynamics via complex variable theory

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Many emerging technologies involving microfluidics, MEMS (micro -electro-mechanical systems) and ``lab-on-chip'' design involve the control and manipulation of fluids at very small scales when the motion and dynamical mechanisms are very different to that of our day-to-day experience of fluids. Viscous, or frictional, effects dominate and lead to a range of new effects that need to be well understood. Such flows are referred to as Stokes flows or low Reynolds number flows. The same class of flows arise in understanding the motion of swimming microorganisms such as spermatozoa or E Coli bacteria. A wide range of experimentally observed behaviour associated with such swimming organisms remains to be properly explained and many of these phenomena have hydrodynamical underpinnings. This research aims to develop simple mathematical models, centred on the use of a powerful set of mathematical techniques from complex analysis, for understanding some of the hydrodynamical mechanisms. In particular, we will focus on how the presence of boundaries - such as no-slip walls where the fluid velocity must vanish or free surfaces on which surface tension is active - can affect the dynamical behaviour of the fluid or the swimming microorganisms.A second component of our project is to study reduced models, again using complex analysis, to understand the dynamics of vorticity when the fluid in which the vorticity is present is compressible. (Sound waves, for example, are a manifestation of a compressible fluid and their interaction with vortical structures associated with aircraft wakes is an important area of study in terms of minimizing noise pollution).