2000/12/31 by V. P. Ruban, D. I. Podolsky, Dmitry Podolsky +2
Engineering · Mathematics · Physics and Astronomy · #Classical mechanics #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory #Gravitational singularity #Instability #Inviscid flow #Mathematical analysis #Mathematical physics #Mathematics #Mechanics #Navier-Stokes equation solutions #Nonlinear system #Physics #Quantum mechanics #Singularity #Vortex #Vortex sheet #Vorticity #physics.flu-dyn
paper · pdf · doi:10.1103/physreve.63.056306
published as Phys. Rev. E, 63, 056306 (2001) · LaTeX, 17 pages, 3 eps figures. This version is close to the journal paper
openalex publication_date 2001/04/16 · arxiv created 2001/04/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L approximately integral k(alpha)/vk/2dk in 3D Fourier representation, where alpha is a constant, 0<alpha<1. Unlike the case alpha=0 (the usual Eulerian hydrodynamics), a finite value of alpha results in a finite energy for a singular, frozen-in vortex filament. This property allows us to study the dynamics of such filaments without the necessity of a regularization procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of antiparallel vortex filaments and an analog of the Crow instability is found at small wave numbers. A local approximate Hamiltonian is obtained for the nonlinear long-scale dynamics of this system. Self-similar solutions of the corresponding equations are found analytically. They describe the formation of a finite time singularity, with all length scales decreasing like (t*-t)(1/(2-alpha)), where t* is the singularity time.