Abstract
This paper considers nonlinear equations describing the propagation of long waves in two-dimensional shear flow of a heavy ideal incompressible fluid with a free boundary. A nine-dimensional group of transformations admitted by the equations of motion is found by symmetry methods. Two-dimensional subgroups are used to find simpler integrodifferential submodels which define classes of exact solutions, some of which are integrated. New steady-state and unsteady rotationally symmetric solutions with a nontrivial velocity distribution along the depth are obtained.
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References
L. V. Ovsyannikov, N. I. Makarenko, V. I. Nalimov, et al., Nonlinear Problems of the Theory of Surface and Internal Waves [in Russian], Nauka, Novosibirsk (1985).
D. J. Benney, “Some properties of long nonlinear waves,” Stud. Appl. Math., 52, 45–50 (1973).
B. E. Zakharov, “Benny equations and quasiclassical approximation in the method of inverse problem,” Funkts. Anal. Prilozh., 14, No. 2, 15–24 (1980).
P. L. Sachdev, “Exact self-similar time-dependent free surface flow under gravity,” J. Fluid Mech., 96, 797–802 (1980).
E. Varley and P. A. Blythe, “Long eddies in sheared flows,” Stud. Appl. Math., 68, 103–187 (1983).
V. M. Teshukov, “On the hyperbolicity of the long-wave equations,” Dokl. Akad. Nauk SSSR, 284, No. 3, 555–562 (1985).
V. Yu. Liapidevskii and V. M. Teshukov, Mathematical Models for Long-Wave Propagation in an Inhomogeneous Fluid [in Russian], Izd. Sib. Otd. Ross. Akad. Nauk, Novosibirsk (2000).
V. M. Teshukov, “Spatial simple waves on a shear flow,” J. Appl. Mech. Tech. Phys., 43, No. 5, 661–670 (2002).
V. M. Teshukov, “Two-dimensional stationary long waves in shear flows,” J. Appl. Mech. Tech. Phys., 45, No. 2, 172–180 (2004).
A. S. Pavlenko, “Symmetry and solutions of the equations of two-dimensional motion of a polytropic gas,” Sib. Élektron. Mat. Izv., 2, 291–307 (2005). (http://semr.math.nsc.ru.)
L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York (1982).
L. V. Ovsyannikov, “On optimal systems of subalgebras,” Dokl. Ross. Akad. Nauk, 333, No. 6, 702–704 (1993).
L. V. Ovsyannikov, “The SUBMODEL program. Gas dynamics,” Prikl. Mat. Mekh., 58, No. 4, 30–55 (1994).
N. I. Muskhelishvili, Singular Integral Equations, Noordhoff, Leyden (1977).
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Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 41–54, September–October, 2008.
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Chesnokov, A.A. Symmetries and exact solutions of the shallow water equations for a two-dimensional shear flow. J Appl Mech Tech Phy 49, 737–748 (2008). https://doi.org/10.1007/s10808-008-0092-5
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DOI: https://doi.org/10.1007/s10808-008-0092-5