Abstract
We prove the existence of steady vortex rings of an ideal fluid in a uniform flow. We use an approach based on a variational principle for the vorticity. We show equally that the maximiser (which represents a quantity related to the vorticity) of a functional related to kinetic energy and the impulse over a class of rearrangements of a prescribed function \(\zeta _0\), is in fact a rearrangement of \(\zeta _0\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambrosetti, A., Struwe, M.: Existence of steady vortex rings in an ideal fluid. Arch. Ration. Mech. Anal. 108, 97–109 (1989)
Badiani, T.V., Burton, G.R.: Vortex rings in \(\mathbb{R}^{3}\) and rearrangements. Proc. Royal Soc. Lond. A. 457, 1115–1135 (2001)
Benjamin, T.B.: The alliance of practical and analytical insight into the nonlinear problems of fluid mechanics. Applications of Methods of Functional Analysis to Problems in Mechanics. Lecture Notes in Mathematics 503, pp. 8-29. Springer, (1976)
Burton, G.R.: Rearrangements of functions, maximization of convex functionals and vortex rings. Math. Ann. 276, 225–253 (1987)
Burton, G.R.: Vortex-rings in a cylinder and rearrangements. J. Diff. Equ. 70, 333–348 (1987)
Burton, G.R.: Vortex-rings of prescribed impulse. Math. Proc. Cambridge Philos. Soc. 134, 515–528 (2003)
Douglas, R.J.: Rearrangements and Nonlinear Analysis of Vortices. PhD thesis, University of Bath (1992)
Douglas, R.J.: Rearrangements of function on unbounded domains. Proc. R. Soc. Edinb. 124A, 621–644 (1994)
Fraenkel, L.E., Berger, M.S.: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 14–51 (1975)
Fraenkel, L.E., Amick, C.J.: The uniqueness of Hill’s spherical vortex. Arch. Ration. Mech. Anal. 92, 91–119 (1986)
Friedman, A., Turkington, B.: Vortex rings: existence and asymptotic estimates. Trans. Am. Math. Soc. 268, 1–37 (1987)
Hill, M.J.M.: On a spherical vortex. Philos. Trans. R. Soc. Lond. A. 185, 213–245 (1894)
Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press, London (1932)
Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, p. 14. AMS, Providence (1997)
Ni, W.M.: On the existence of global vortex rings. J. d’Anal. Math. 37, 208–247 (1980)
Norbury, J.: A steady vortex ring close to Hill’s spherical vortex. Proc. Cambridge Philos. Soc. 72, 253–284 (1972)
Rebah, D.: A steady vortex ring in Poiseuille flow and rearrangements of function. Proc R. Soc. Lond. A 462, 1235–1253 (2006)
Rebah, D.: Steady vortex pairs in two phase shear flow of an ideal fluid. J. Math. Anal. Appl. 317, 14–27 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Rebah, D. Steady Vortex Rings in a Uniform Flow and Rearrangements of a Function. Results Math 75, 23 (2020). https://doi.org/10.1007/s00025-019-1148-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-019-1148-y