Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-01T21:28:46.783Z Has data issue: false hasContentIssue false
  • English
  • Français

A vortex filament moving without change of form

Published online by Cambridge University Press:  20 April 2006

Shigeo Kida
Shigeo Kida
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

The motion of a very thin vortex filament is investigated using the localized induction equation. A family of vortex filaments which move without change of form are obtained. They are expressed in terms of elliptic integrals of the first, second and third kinds. In general they do not close and have infinite lengths. In some particular cases they take the form of closed coils which wind a doughnut. There exist a family of closed vortex filaments which do not travel in space but only rotate around a fixed axis. Our solutions include various well-known shapes such as the circular vortex ring, the helicoidal filament, the plane sinusoidal filament, Euler's elastica and the solitary-wave-type filament. It is shown that they correspond to the travelling wave solution of a nonlinear Schrödinger equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 112 , November 1981 , pp. 397 - 409
Copyright
© 1981 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Chap. 16–17. Dover.
Asano, N., Taniuti, T. & Yajima, N. 1969 Perturbation method for a nonlinear wave modulation. II. J. Math. Phys. 10, 2020–2024.Google Scholar
Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. J. Math. & Phys. 46, 133–139.Google Scholar
Bespalov, V. I. & Talanov, V. I. 1966 Filamentary structure of light beams in nonlinear liquids. J. Exp. Theor. Phys. Lett. 3, 307–310.Google Scholar
Betchov, R. 1965 On the curvature and torsion of an isolated vortex filament. J. Fluid Mech. 22, 471–479.Google Scholar
Eisenhart, L. P. 1960 A Treatise on the Differential Geometry of Curves and Surfaces, 13–15. Dover.
Fraenkel, L. E. 1970 On steady vortex rings of small cross-section in an ideal fluid. Proc. Roy. Soc. A 316, 29–62.Google Scholar
Fraenkel, L. E. 1972 Examples of steady vortex rings of small cross-section in an ideal fluid. J. Fluid Mech. 51, 119–135.Google Scholar
Hama, F. R. 1962 Progressive deformation of a curved vortex filament by its own induction. Phys. Fluids 5, 1156–1162.Google Scholar
Hama, F. R. 1963 Progressive deformation of a perturbed line vortex filament. Phys. Fluids 6, 526–534.Google Scholar
Hasimoto, H. 1971 Motion of a vortex filament and its relation to elastica. J. Phys. Soc. Japan 31, 293–294.Google Scholar
Hasimoto, H. 1972 A soliton on a vortex filament. J. Fluid Mech. 51, 477–485.Google Scholar
Hasimoto, H. & Ono, H. 1972 Nonlinear modulation of gravity waves. J. Phys. Soc. Japan 33, 805–811.Google Scholar
Kambe, T. & Takao, T. 1971 Motion of distorted vortex rings. J. Phys. Soc. Japan 31, 591–599.Google Scholar
Karpman, V. I. & Krushkal, E. M. 1969 Modulated waves in nonlinear dispersive media. Sov. Phys. J. Exp. Theor. Phys. 28, 277–281.Google Scholar
Kelley, P. L. 1965 Self-focusing of optical beams. Phys. Rev. Lett. 15, 1005–1008.Google Scholar
Kelvin, Lord 1880 Mathematical and Physical Papers, vol. 4, p. 152. Cambridge University Press.
Lakshmanan, M. 1977 Continuum spin system as an exactly solvable dynamical system. Phys. Lett. A 61, 53–54.Google Scholar
Lakshmanan, M., Ruijgrok, T. W. & Thompson, C. J. 1976 On the dynamics of a continuum spin system. Physica A 84, 577–590.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Lamb, G. L. 1977 Solitons on moving space curves. J. Math. Phys. 18, 1654–1661.Google Scholar
Lighthill, M. J. 1965 Contributions to the theory of waves in non-linear dispersive systems. J. Inst. Math. Applic. 1, 269–306.Google Scholar
Lighthill, M. J. 1967 Some special cases treated by the Whitham theory. Proc. Roy. Soc. A 299, 28–53.Google Scholar
Love, A. E. H. 1927 A Treatise on the Mathematical Theory of Elasticity, 4th edn, p. 401. Cambridge University Press.
Moore, D. W. & Saffman, P. G. 1972 The motion of a vortex filament with axial flow. Phil. Trans. Roy. Soc. A 272, 403–429.Google Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento Suppl. 6, 279–287.Google Scholar
Saffman, P. G. 1961 Propagation of a solitary wave along a magnetic field in a cold collision-free plasma. J. Fluid Mech. 11, 16–20.Google Scholar
Saffman, P. G. 1970 The velocity of viscous vortex rings. Stud. Appl. Math. 49, 371–380.Google Scholar
Scott, A. C., Chu, F. Y. F. & Mclaughlin, D. W. 1973 The soliton: A new concept in applied science. Proc. I.E.E.E. 61, 1443–1483.Google Scholar
Talanov, V. I. 1965 Self focusing of wave beams in nonlinear media. J. Exp. Theor. Phys. Lett. 2, 138–141.Google Scholar
Taniuti, T. & Yajima, N. 1969 Perturbation method for a nonlinear wave modulation. I. J. Math. Phys. 10, 1369–1372.Google Scholar
Tung, C. & Ting, L. 1967 Motion and decay of a vortex ring. Phys. Fluids 10, 901–910.Google Scholar
Widnall, S. E. 1972 The stability of a helical vortex filament. J. Fluid Mech. 54, 641–663.Google Scholar
Zakharov, V. E. & Manakov, S. V. 1974 On the complete integrability of a nonlinear Schrödinger equation. Theor. Math. Phys. 19, 551–559.Google Scholar
Zakharov, V. E. & Shabat, A. B. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. J. Exp. Theor. Phys. 34, 62–69.Google Scholar
Zakharov, V. E. & Shabat, A. B. 1973 Interaction between solitons in a stable medium. Sov. Phys. J. Exp. Theor. Phys. 37, 823–828.Google Scholar