Geodesic
Discrete Spectrum of Bifurcation of Exact Solutions for Stationary Longitudinal Waves in the Flow of Perfect Fluid Around a Circular Body of a Large Radius
R. N. Ibragimov1
1GE Global Research 1 Research Circle Niskayuna, NY 12309, USA
Mathematical modelling of natural phenomena, Tome 11 (2016) no. 2, pp. 63-74

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We consider a free boundary problem for incompressible perfect fluid circulating around a circle Γ of a large radius, i.e. within a central gravity field. The outward curve γ is a free boundary to be sought. We assume that the flow, which is confined between Γ and γ, is irrotational. The centrifugal force caused by the circulation of the flow makes the fluid go outward. We show that there exist stationary waves, which are periodic oscillations of the fluid and are exact solutions corresponding to bifurcating branches emanating from trivial solution. In the frame of the problem in question, the wave number can take only the integer multiplies of 2π. This makes the spectrum discrete, which makes a crucial difference with the problem for flows in an infinite domain over the straight line or plane (i.e. the ocean problem). To prove the existence of exact solutions, the method of conformal mapping is used. By this device the free boundary problem is transformed into a boundary value problem in a fixed domain.
DOI : 10.1051/mmnp/201611205

R. N. Ibragimov  1

1 GE Global Research 1 Research Circle Niskayuna, NY 12309, USA 
R. N. Ibragimov. Discrete Spectrum of Bifurcation of Exact Solutions for Stationary Longitudinal Waves in the Flow of Perfect Fluid Around a Circular Body of a Large Radius. Mathematical modelling of natural phenomena, Tome 11 (2016) no. 2, pp. 63-74. doi: 10.1051/mmnp/201611205
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[1] A. Aguiar, P. Read, R. Wordsworth, T. Salter, Y. Yamazaki Icarus 2010 755 763

[2] R. Anderson, S. Ali, L. Brandtmiller, S. Nielsen, M. Fleisher Science 2009 1443 1449

[3] G. Bachelor. An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1967.

[4] J. Beal Comm. Pure. Appl. Math. 1991 211 257

[5] J. Bell. Hubble captures best view of mars ever obtained From Earth. HubbleSite. NASA. http://hubblesite.org/newscenter/archive/releases/2001/24, (2001), Retrieved 2010-02-27.

[6] G. Ben-Yu Math. Comput. 1995 1067

[7] E. Blinova. A hydrodynamical theory of pressure and temperature waves and of centers of atmospheric action. C. R. Dokl. Acad. Sci.URSS 39, 257, 1943.

[8] E. Blinova. A method of solution of the nonlinear problem of atmospheric motions on a planetary scale. Dokl. Akad. Nauk SSSR N.S.110, 975, 1956 .

[9] M. Boehm, S. Lee J. Atmos. Sci. 2003 247 261

[10] K. Friedrichs, D. Hyers. The existence of solitary waves. Comm. Pure. Appl. Math. 7, 1954.

[11] R. Gardner J. Differential Equations 1982 343 364

[12] N. Ibragimov, R. Ibragimov Phys. Lett. A. 2011 3858 3865

[13] R. Ibragimov Phys. Fluids 2011 123102

[14] R. Ibragimov, G. Jefferson, J. Carminati Int. J. Non-Linear Mech. 2013 28 44

[15] R. Ibragimov, G. Jefferson, J. Carminati Springer: Anal. Math. Phys. 2013 201 294

[16] R. Ibragimov, D. Pelinovsky J. Math. Fluid Mech. 2009 60

[17] R. Ibragimov, H. Villasenor J. Fluids Eng.

[18] R. Ibragimov, N. Ibragimov, L. Galiakberova Math. Model. Nat. Phenom. 2014 32 39

[19] R. Ibragimov, L. Guang Dyn. Atmos. Oceans Volume 2015 1 11

[20] R. Ibragimov Math. Phys. Anal. Geom. 2001 51 53

[21] D. Iftimie, G. Raugel, G. Some results on the NS equations in thin three-dimensional domains. J. Differ. Equations 169, 281, 2001.

[22] T. Levi-Civita Math. Ann. 1925 256 314

[23] L. Lions, R. Teman, S. Wang. S. On the equations of the large-scale ocea., Nonlinearity 5, 1007, 1992.

[24] L. Lions, R. Teman, S. Wang. New formulations of the primitive equations of atmosphere and applications. Nonlinearity 5, 237, 1992.

[25] Nekrasov, A.I., 1952: Exact theory of the uniform waves on the surface of hard liquid, USSR Academy of Science, Moscow.

[26] H. Okamoto J. Math. Soc. Japan 1986

[27] C. Summerhayes, S Thorpe. Oceanography: An Illustrative Guide. Wiley, New York, 1996.

[28] W. Weijer, F. Vivier, S. Gille, H. Dijkstra, H. Multiple oscillatory modes of the Argentine Basin. Part II: The spectral origin of basin modes. J. Phys. Oceanogr. 37, 2869, 2007.

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