Asymptotic description of vortex filaments in an incompressible fluid

✍ Scribed by A.I Shafarevich

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 810 KB
• Volume: 64
• Category: Article
• ISSN: 0021-8928
• DOI: 10.1016/s0021-8928(00)00047-2

✦ Synopsis

General equations which descrii vortex filaments in an incompressible, low viscosity fluid are derived. Perturbation theory, which leads to these equations, is based on the topological properties of the trajectories of steady flows of an ideal fluid. The vortex filament equations turn out to be similar to the Prandtl equations in boundary-layer theory and the equations of V. I? Ma&v, which describe periodic "coherent" pulsations. An infinite series of integral relations, which, when the viscosity disappears, reduce to conservation laws, is found in the case of these equations. It is shown that the well known Moffatt-Kida-Ohkitani vortex is the simplest "steady" solution of the equations of a vortex filament. The equations of an elongated vortex are naturally treated as equations which are defined in a graph with boundary conditions at its vertices. The graph which occurs in this case is found to be associated in a natural way with the Morse theory and the topological theory of integrable Hamiltonian systems and is identical to the Reef and Fomenko invariants which are well known in these theories.