JP Journal of Geometry and Topology
Volume 8, Issue 2, Pages 151 - 184
(July 2008)
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TOPOLOGICAL IDEAS AND STRUCTURES IN FLUID DYNAMICS
Luciano Boi (France)
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Abstract:
The realization that the mathematical disciplines of topology and geometry are extremely valuable in furthering our understanding of fluid flows has been evolving steadily for many years within the fluid dynamics community and in other related fields. Since Poincaré’s seminal work, it has been recognised that the global geometric point of view is essential for understanding Newtonian mechanics. Modern differential geometry and topology have brought new insight into fluid dynamical systems having symmetry and geometrical structures. Integrals of motions or conservation laws were found to be closely related to the structure of the topological space. The existence of certain integrals of the equations of magnetohydrodynamics (MHD) and inviscid hydrodynamics, for example, can be interpreted in terms of the linking and twisting of magnetic fields and vortex lines. In this article, we focus on the relationship between some knot and link invariants such as the Jones-Witten invariants and certain invariants for vector fields in fluid mechanics. We stress the fact that topological properties are suited for describing and explaining many physical and dynamical characteristics of fluids and other striking physical phenomena. |
| Keywords and phrases: topology of knots, writhing number, linking number, helicity and invariants, fluid flows, dynamics, magnetic fields, braids, reconnection, energy of knots and links, topological changes, dissipation. |
| Communicated by Yasuo Matsushita |
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