E-Book Overview
This monograph aims at giving a presentation of recent and new ideas that arise from the problems of planar fluid dynamics and which are interesting from the point of view of geometric function theory and potential theory. In particular, this book is concerned with geometric problems for Hele-Shaw flows. Also Hele-Shaw flows on parameter spaces (e.g., the Teichmüller space) are treated and connections with string theory are revealed. Ultimately, the interaction between several branches of complex and potential analysis, and planar fluid mechanics is discussed. For most parts of this book the background provided by graduate courses in real and complex analysis, in particular, the theory of conformal mappings and in fluid mechanics is assumed. There are some historical remarks concerning the people that have contributed to the topic. The book is as self-contained as possible.
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Bj¨orn Gustafsson Alexander Vasil’ev Conformal and Potential Analysis in Hele-Shaw cells Stockholm-Valpara´ıso July 2004 Preface One of the most influential works in Fluid Dynamics at the edge of the 19th century was a short paper [130] written by Henry Selby Hele-Shaw (1854–1941). There Hele-Shaw first described his famous cell that became a subject of deep investigation only more than 50 years later. A Hele-Shaw cell is a device for investigating two-dimensional flow of a viscous fluid in a narrow gap between two parallel plates. This cell is the simplest system in which multi-dimensional convection is present. Probably the most important characteristic of flows in such a cell is that when the Reynolds number based on gap width is sufficiently small, the Navier-Stokes equations averaged over the gap reduce to a linear relation similar to Darcy’s law and then to a Laplace equation for pressure. Different driving mechanisms can be considered, such as