Leading experts present a unique, invaluable introduction to the study of the geometry and typology of fluid flows. From basic motions on curves and surfaces to the recent developments in knots and links, the reader is gradually led to explore the fascinating world of geometric and topological fluid mechanics. Geodesics and chaotic orbits, magnetic knots and vortex links, continual flows and singularities become alive with more than 160 figures and examples. In the opening article, H. K. Moffatt sets the pace, proposing eight outstanding problems for the 21st century. The book goes on to provide concepts and techniques for tackling these and many other interesting open problems.
Table of ContentsPreface. Photograph of H.K. Moffatt. I: Eight Problems for the XXI Century. Some Remarks on Topological Fluid Mechanics; H.K. Moffatt. II: Mathematics Background. Differential Geometry of Curves and Surfaces; R. Langevin. Topology in Four Days; T. Tokieda. Elements of Classical Knot Theory; C. Weber. An Introduction to Knot Theory; L.H. Kauffman. Fluid Mechanics and Mathematical Structures; P. Boyland. III: Geometry and Topology of Fluid Flows. Introduction to a Geometrical Theory of Fluid Flows and Dynamical Systems; T. Kambe. Streamline Patterns and their Bifurcations Using Methods from Dynamical Systems; M. Brøns. Topological Features of Inviscid Flows; R. Grhist, R. Komendarczyk. Geometric and Topological Aspects of Vortex Motion; R.L. Ricca. Topology Bounds the Energy; B.A. Khesin. Measures of Topological Structure in Magnetic Fields; M.A. Berger. Diffeomorphisms, Braids and Flows; A. Shnirelman. Variational Principles, Geometry and Topology of Lagrangian-Averaged Fluid Dynamics; D.D. Holm. IV: Reconnections and Singularities. The Geometry of Reconnection; G. Hornig. Euler Singularities from the Lagrangian Viewpoint; S. Childress. Analysis of a Candidate Flow for Hydrodynamic Blowup; R.B. Pelz. Subject Index.