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Finite time singularities in a class of hydro dynamic models

  • Victor P. Ruban
  • Dmitry I. Podolsky
  • Jens J. Rasmussen
Conference paper
Part of the Fluid Mechanics and Its Applications book series (FMIA, volume 71)

Abstract

Models of inviscid incompressible fluid are considered, with the kinetic energy (i.e., the Lagrangian functional) taking the form L∼∫k α|vk|2 d 3k in 3D Fourier representation, where α is a constant, 0“;α<1. Unlike the case α=0 (the usual Eulerian hydrodynamics), a finite value of & ga 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 regularisation procedure for short length scales. The linear analysis of small symmetrical deviations from a stationary solution is performed for a pair of anti-parallel 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−α) where t* is the singularity time.

Keywords

Vortex Line Vortex Filament Hydrodynamic Type Vorticity Maximum Short Length Scale 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Kuznetsov, E.A. & Ruban, V.P. 2000 Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems. Phys. Rev. E61, 831.MathSciNetADSCrossRefGoogle Scholar
  2. Ruban, V.P. 1999 Motion of magnetic flux lines in magnetohydrodynamics. JETP89, 299.CrossRefADSGoogle Scholar
  3. Ruban, V.P., Podolsky, D.I., & Rasmussen, J.J. 2001 Finite time singularities in a class of hydrodynamic models. Phys. Rev. E63, 056306.Google Scholar
  4. Ruban, V.P. 2001 Slow inviscid flows of a compressible fluid in spatially inhomogeneous systems. Phys. Rev. E64, 036305.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Victor P. Ruban
    • 1
    • 2
  • Dmitry I. Podolsky
    • 1
  • Jens J. Rasmussen
    • 2
  1. 1.L. D. Landau Institute for Theoretical PhysicsMoscowRussia
  2. 2.Optics and Fluid Dynamics DepartmentOFD-129, Risø National LaboratoryRoskildeDenmark

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