On The Motion of Three Vortices

J. L. Synge

doi:10.4153/CJM-1949-022-2

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In a perfect incompressible fluid extending to infinity, the determination of the motion of N parallel rectilinear vortex filaments involves the solution of N non-linear differential equations, each of the first order. The method of Kirchhoff provides certain constants of the motion. If we describe the positions of the vortices by their point-traces on a plane perpendicular to them, the following facts follow from the theory of Kirchhoff:

(1.1) The mean centre of the system is fixed.

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Lakshmana Rao, S. K. 1951. Fixed configurations of four rectilinear vortex filaments. Proceedings of the Indian Academy of Sciences - Section A, Vol. 34, Issue. 4,

Lakshmana Rao, S. K. 1953. The motion of four rectilinear vortex filaments. Proceedings of the Indian Academy of Sciences - Section A, Vol. 38, Issue. 2, p. 143.

Marsden, Jerrold and Weinstein, Alan 1983. Coadjoint orbits, vortices, and Clebsch variables for incompressible fluids. Physica D: Nonlinear Phenomena, Vol. 7, Issue. 1-3, p. 305.

Aref, Hassan 1985. Theoretical and Applied Mechanics. p. 43.

Kimura, Y. and Hasimoto, H. 1985. Dynamical Problems in Soliton Systems. Vol. 30, Issue. , p. 164.

O’Neil, Kevin Anthony 1987. Stationary configurations of point vortices. Transactions of the American Mathematical Society, Vol. 302, Issue. 2, p. 383.

Kimura, Yoshifumi 1987. Similarity Solution of Two-Dimensional Point Vortices. Journal of the Physical Society of Japan, Vol. 56, Issue. 6, p. 2024.

Tavantzis, John and Ting, Lu 1988. The dynamics of three vortices revisited. The Physics of Fluids, Vol. 31, Issue. 6, p. 1392.

Lim, Chjan C. 1988. Singular manifolds and quasi-periodic solutions of Hamiltonians for vortex lattices. Physica D: Nonlinear Phenomena, Vol. 30, Issue. 3, p. 343.

Aref, Hassan 1989. Three-vortex motion with zero total circulation: Addendum. ZAMP Zeitschrift f�r angewandte Mathematik und Physik, Vol. 40, Issue. 4, p. 495.

Aref, Hassan 1992. Studies in Turbulence. p. 568.

Omoto, Hideo and Kimura, Yoshifumi 1992. On ε-approximate singularities of autonomous systems of vortex type. Nagoya Mathematical Journal, Vol. 125, Issue. , p. 15.

Meleshko, V. V. Konstantinov, M. Yu. Gurzhi, A. A. and Konovaljuk, T. P. 1992. Advection of a vortex pair atmosphere in a velocity field of point vortices*. Physics of Fluids A: Fluid Dynamics, Vol. 4, Issue. 12, p. 2779.

Konstantinov, M. 1994. Chaotic phenomena in the interaction of vortex rings. Physics of Fluids, Vol. 6, Issue. 5, p. 1752.

Newton, Paul K. 1994. Hannay-Berry phase and the restricted three-vortex problem. Physica D: Nonlinear Phenomena, Vol. 79, Issue. 2-4, p. 416.

Kuznetsov, Leonid and Zaslavsky, George M. 1998. Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas. Vol. 511, Issue. , p. 199.

Ting, L. Klein, R. and Knio, O. M. 1998. IUTAM Symposium on Dynamics of Slender Vortices. Vol. 44, Issue. , p. 3.

Kuznetsov, Leonid and Zaslavsky, George M. 1998. Regular and chaotic advection in the flow field of a three-vortex system. Physical Review E, Vol. 58, Issue. 6, p. 7330.

Pekarsky, Sergey and Marsden, Jerrold E. 1998. Point vortices on a sphere: Stability of relative equilibria. Journal of Mathematical Physics, Vol. 39, Issue. 11, p. 5894.

Kidambi, Rangachari and Newton, Paul K. 1998. Motion of three point vortices on a sphere. Physica D: Nonlinear Phenomena, Vol. 116, Issue. 1-2, p. 143.

Download full list

Article contents

On The Motion of Three Vortices

Extract

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests