Reduced models in fluid dynamics via complex variable theory

许多涉及微流体、微机电系统（MEMS）和“芯片实验室”设计的新兴技术涉及在非常小的尺度上控制和操纵流体，当运动和动力机制与我们日常的流体经验非常不同时。粘性或摩擦效应占主导地位，并导致一系列需要充分理解的新效应。这样的流称为斯托克斯流或低雷诺数流。在理解游动的微生物（如精子或大肠杆菌）的运动时，也出现了同样的流动。与这些游泳生物相关的大量实验观察行为仍有待适当解释，其中许多现象具有流体动力学基础。本研究旨在开发简单的数学模型，集中使用一套强大的复杂分析数学技术，以理解一些流体动力机制。特别是，我们将重点关注边界的存在如何影响流体或游动微生物的动力学行为，例如流体速度必须消失的防滑壁或表面张力活跃的自由表面。我们项目的第二个组成部分是研究简化模型，同样使用复杂分析，以理解当存在涡度的流体是可压缩时涡度的动力学。（例如，声波是可压缩流体的一种表现形式，它们与飞机尾迹相关的垂直结构的相互作用是最小化噪声污染方面的一个重要研究领域）。

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).