International Journal of Differential Equations

Some Nonlinear Vortex Solutions

Michael C. Haslam, Christopher J. Smith, Ghada Alobaidi, and Roland Mallier

Full-text: Open access

Abstract

We consider the steady-state two-dimensional motion of an inviscid incompressible fluid which obeys a nonlinear Poisson equation. By seeking solutions of a specific form, we arrive at some interesting new nonlinear vortex solutions.

Article information

Source
Int. J. Differ. Equ., Volume 2012 (2012), Article ID 929626, 10 pages.

Dates
Received: 3 June 2011
Revised: 10 November 2011
Accepted: 24 November 2011
First available in Project Euclid: 24 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485226818

Digital Object Identifier
doi:10.1155/2012/929626

Mathematical Reviews number (MathSciNet)
MR2898145

Zentralblatt MATH identifier
06078092

Citation

Haslam, Michael C.; Smith, Christopher J.; Alobaidi, Ghada; Mallier, Roland. Some Nonlinear Vortex Solutions. Int. J. Differ. Equ. 2012 (2012), Article ID 929626, 10 pages. doi:10.1155/2012/929626. https://projecteuclid.org/euclid.ijde/1485226818


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References

  • J.-M. Chomaz, “Global instabilities in spatially developing flows: non-normality and nonlinearity,” Annual Review of Fluid Mechanics, vol. 37, pp. 357–392, 2005.
  • P. G. Saffman, Vortex Dynamics, Cambridge University Press, New York, NY, USA, 1992.
  • J. M. Faddy and D. I. Pullin, “Flow structure in a model of aircraft trailing vortices,” Physics of Fluids, vol. 17, no. 8, Article ID 085106, 2005.
  • T. Siegmann-Hegerfeld, S. Albensoeder, and H. C. Kuhlmann, “Two- and three-dimensional flows in nearly rectangular cavities driven by collinear motion of two facing walls,” Experiments in Fluids, vol. 45, no. 5, pp. 781–796, 2008.
  • K. W. Chow, S. C. Tsang, and C. C. Mak, “Another exact solution for two-dimensional, inviscid sinh Poisson vortex arrays,” Physics of Fluids, vol. 15, no. 8, pp. 2437–2440, 2003.
  • A. Tur, V. Yanovsky, and K. Kulik, “Vortex structures with complex points singularities in two-dimensional Euler equations. New exact solutions,” Physica D, vol. 240, no. 13, pp. 1069–1079, 2011.
  • J. T. Stuart, “On finite amplitude oscillations in laminar mixing layers,” Journal of Fluid Mechanics, vol. 29, pp. 417–440, 1967.
  • R. Mallier and S. A. Maslowe, “A row of counter-rotating vortices,” Physics of Fluids A, vol. 5, no. 4, pp. 1074–1075, 1993.
  • J. Liouville, “Sur l'équation aux différences partielles ${\partial }^{2}\text{log}\lambda /\partial u\partial v\pm {\lambda }^{2}/2{a}^{2}=0$,” Journal de Mathématiques Pures et Appliquées, vol. 18, pp. 71–72, 1853.
  • D. G. Crowdy, “General solutions to the 2D Liouville equation,” International Journal of Engineering Science, vol. 35, no. 2, pp. 141–149, 1997.
  • J. Schmid-Burgk, Zweidimensionale selbstkonsistente Lösungen stationären Wlassovgleichung für Zweikomponentenplasmen, Diplomarbeit, Ludwig-Maximilians-Universität München, 1965.
  • G. W. Walker, “Some problems illustrating the forms of nebuloe,” Proceedings of the Royal Society A, vol. 91, no. 631, pp. 410–420, 1915.
  • A. Barcilon and P. G. Drazin, “Nonlinear waves of vorticity,” Studies in Applied Mathematics, vol. 106, no. 4, pp. 437–479, 2001.
  • A. C. Ting, H. H. Chen, and Y. C. Lee, “Exact solutions of a nonlinear boundary value problem: the vortices of the two-dimensional sinh-Poisson equation,” Physica D, vol. 26, no. 1–3, pp. 37–66, 1987.
  • M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications Inc., New York, NY, USA, 1992.