Hostname: page-component-8448b6f56d-c47g7 Total loading time: 0 Render date: 2024-04-23T09:49:17.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic estimates for symmetric vortex streets

Published online by Cambridge University Press:  17 February 2009

G. Keady
G. Keady
Affiliation:
Mathematics Department, University of Western Australia, Nedlands, W.A. 6009.
Rights & Permissions [Opens in a new window]

Abstract

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.

Steady plane inviscid symmetric vortex streets are flows defined in the strip R × (0, b) and periodic in x with period 2a in which the flow in (−a, a) × (0, b) is irrotational outside a vortex core on which the vorticity takes a prescribed constant value. A family of such vortex street flows, characterised by a variational principle in which the area |Aα| and the centroid yc of the vortex core Aα are fixed, will be considered. For such a family, indexed by a parameter α, suppose that the cores Aα become small in the sense that

Asymptotic estimates on functionals such as flux constant and speed are obtained.

Type
Research Article
Information
The ANZIAM Journal , Volume 26 , Issue 4 , April 1985 , pp. 487 - 502
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Benjamin, T. B., “The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics”, in Applications of methods of functional analysis to problems in mechanics, Symposium IUTAM/IMU Marseilles Sept. 1975 (eds. Germain, P. and Nayroles, B.), Lecture Notes in Math. 503 (Springer, Berlin, 1976), 8–29.Google Scholar
[2]Berger, M. S. and Fraenkel, L. E., “Nonlinear desingularization in certain free-boundary problem”, Comm. Math. Phys. 77 (1980), 149–172.CrossRefGoogle Scholar
[3]Caffarelli, L. A. and Friedman, A., “Asymptotic estimates for the plasma problem”, Duke Math. J. 47 (1980), 705–762.CrossRefGoogle Scholar
[4]Fraenkel, L. E., “A lower bound for electrostatic capacity in the plane”, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 267–273.Google Scholar
[5]Fraenkel, L. E. and Berger, M. S., “A global theory of steady vortex rings in an ideal fluid”, Acta Math. 132(1974), 13–51.CrossRefGoogle Scholar
[6]Friedman, A. and Turkington, B., “Vortex rings: existence and asymptotic estimates”, Trans. Amer. Math. Soc. 268 (1981), 1–38.CrossRefGoogle Scholar
[7]Gidas, B., Ni, W. and Nirenberg, L., “Symmetry via the maximum principle”, Comm. Math. Phys. 68(1979), 209–243.CrossRefGoogle Scholar
[8]Keady, G. and Norbury, J., “A semilinear elliptic eigenvalue problem, part II”, Proc. Roy. Soc. Edinburgh Sect. A 87 (1980), 83–109.Google Scholar
[9]Keady, G., “An elliptic boundary-value problem with a discontinuous nonlinearity”, Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 161–174.Google Scholar
[10]Keady, G., “Existence and asymptotic estimates for symmetric vortex streets”, Res. Rep. Math. Dep. Univ. Western Australia (01 1983).Google Scholar
[11]Lamb, H., Hydrodynamics (Cambridge University Press, 6th edition, 1932).Google Scholar
[12]Lichtenstein, L., “Uber einige Existenzprobleme der Hydrodynamik”, Math. Z. 23 (1925), 89–154.CrossRefGoogle Scholar
[13]Lichtenstein, L.. Grundlagen der hydrodynamik (Springer, Berlin, 1929).Google Scholar
[14]McLeon, B., “Rearrangements”, preprint (1983).Google Scholar
[15]Moore, D. W. and Saffman, P. G., “Structure of a line vortex in an imposed strain”, in Aircraft wake turbulence (eds. Olsen, J., Goldberg, A. and Rogers, M.), (Plenum Press, New York, 1971), 339–354.CrossRefGoogle Scholar
[16]Payne, L. E., “Isoperimetric inequalities and their application”, SIAM Rev. 9 (1967), 453–458.CrossRefGoogle Scholar
[17]Pierrehumbert, R. T. and Widnall, S. E., “The structure of organised vortices in a free shear layer”, J. Fluid Mech. 102 (1981), 301–314.CrossRefGoogle Scholar
[18]Polya, G. and Szego, G., Isoperimetric inequalities in mathematical physics (Princeton Univ. Press, 1951).Google Scholar
[19]Rosenhead, L.. “The Karman street of vortices in a channel of finite breadth”, Philos. Trans. Roy. Soc. London Ser. A 228 (1929), 275–329.Google Scholar
[20]Saffman, P. G., “The approach of a vortex pair to a plane surface in an inviscid flow”, J. Fluid Mech. 92 (1979), 497–503.CrossRefGoogle Scholar
[21]Saffman, P. G. and Szeto, R., “Structure of a linear array of uniform vortices”, Stud. Appl. Math. 65 (1981), 223–248.CrossRefGoogle Scholar
[22]Turkington, B., “On steady vortex flow in two dimensions, I”, Comm. Partial Differential Equations 8 (1983), 999–1030.CrossRefGoogle Scholar
[23]Turkington, B., “On steady vortex flow in two dimensions, II”, Comm. Partial Differential Equations 8(1983), 1031–1071.CrossRefGoogle Scholar