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On the existence of solitary waves in rotating fluids

Published online by Cambridge University Press:  14 November 2011

S. M. Sun
S. M. Sun
Affiliation:
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A
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Extract

This paper considers the existence of axisymmetric solitary waves in an inviscid and incompressible rotating fluid bounded by a rigid cylinder. It has been obtained by many experiments and formal derivations that this flow has internal solitary waves in the fluid when equilibrium state at infinity satisfies certain conditions. This paper gives a rigorous proof of the existence of solitary wave solutions for the exact equations governing the flow under such conditions at infinity, and shows that the first-order approximations of the solitary wave solutions for the exact equations are solitary wave solutions derived formally using long-wave approximation. The ideas in the proof of the existence of solitary waves in two-dimensional stratified fluids are used and a main difficulty from the singularity at axis of rotation is overcome.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 1105 - 1129
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Benjamin, T. B.. Theory of the vortex breakdown phenomenon. J. Fluid Mech. 14 (1962), 593–629.CrossRefGoogle Scholar
2Benjamin, T. B.. Some developments in the theory of vortex breakdown. J. Fluid Mech. 28 (1967), 65–84.CrossRefGoogle Scholar
3Benney, D. J.. Long non-linear waves in fluid flows. J. Math. Phys. 45 (1966), 52–63.CrossRefGoogle Scholar
4Berger, M. S.. Remarks on vortex breakdown. In Mathematical Aspects of Vortex Dynamics, ed. Caflisch, R. E., 171–82 (Philadelphia, PA: SIAM, 1989).Google Scholar
5Friedrichs, K. and Hyers, D.. The existence of solitary waves. Comm. Pure Appl. Math. 3 (1954), 517–50.CrossRefGoogle Scholar
6Gilbarg, D. and Trudinger, N. S.. Elliptic Partial Differential Equations of Second Order (Berlin: Springer, 1977).CrossRefGoogle Scholar
7Greenspan, H. P.. The Theory of Rotating Fluids (London: Cambridge University Press, 1968).Google Scholar
8Grimshaw, R. H. J.. Resonant flow of a rotating fluid past an obstacle: The general case. Stud. Appl. Math. 83(1990), 249–69.CrossRefGoogle Scholar
9Harvey, J. K.. Some observations of the vortex breakdown phenomenon. J. Fluid Mech. 14 (1962), 585–92.CrossRefGoogle Scholar
10Howard, L. N. and Gupta, A. S.. On the hydrodynamic and hydromagnetic stability of swirling flows. J. Fluid Mech. 14 (1962), 463–76.CrossRefGoogle Scholar
11Leibovich, S.. Weakly non-linear waves in rotating fluids. J. Fluid Mech. 42 (1970), 803–922.CrossRefGoogle Scholar
12Leibovich, S.. The structure of vortex breakdown. Ann. Rev. Fluid Mech. 10 (1978), 221–46.CrossRefGoogle Scholar
13Long, R. R.. Steady motion around a symmetrical obstacle moving along the axis of a rotating liquid. J. Meteorol. 10 (1953), 197–202.2.0.CO;2>CrossRefGoogle Scholar
14Ni, W. M.. On the existence of global vortex rings. J. Anal. Math. 37 (1980), 208–47.CrossRefGoogle Scholar
15Pritchard, W. G.. Solitary waves in rotating fluids. J. Fluid Mech. 42 (1970), 61–83.CrossRefGoogle Scholar
16Sun, S. M. and Shen, M. C.. A new solitary wave solution for water waves with surface tension. Ann. Mat. Pura Appl. 162 (1992), 179–214.CrossRefGoogle Scholar
17Sun, S. M. and Shen, M. C.. Exact theory of solitary waves in a stratified fluid with surface tension, Part I. Nonoscillatory case. J. Differential Equations 105 (1993), 94–116.CrossRefGoogle Scholar
18Sun, S. M. and Shen, M. C.. Solitary waves in a two-layer fluid with surface tension. SIAM J. Math. Anal. 24(1993), 866–91.CrossRefGoogle Scholar
19Sun, S. M. and Shen, M. C.. Existence of solitary pressure pulses in a cylindrical fluid-filled elastic tube. J. Differential Equations 115 (1995), 224–560.CrossRefGoogle Scholar
20Ter-Krikorov, A. M.. Théorie exacte des ondes longues stationnaires dans un liquide hétérogéne. J. Mécanique 2 (1963), 351–75.Google Scholar
21Titchmarsh, E. C.. Eigenfunction Expansions Associated with Second-Order Differential Equations, Parts I and II, 2nd edn (Oxford: Clarendon Press, 1962).Google Scholar
22Watson, G. N.. A Treatise on the Theory of Bessel Functions, 2nd edn (New York: MacMillan, 1945).Google Scholar
23Yih, C. S., O'Dell, W. and Debler, W. R.. Prevention of stagnation zones in flows of a stratified or a rotating fluid. Proceedings of the 4th U.S. National Congress of Applied Mechanics (1962), 1141–53 (New York: American Society of Mechanical Engineers, 1962).Google Scholar