This paper investigates isobaric motions for which the values of the pressure are conserved in fluid particles. In it, a new analytic exact particular solution of nonlinear multidimensional hydrodynamic equations is obtained; it describes a trochoidal wave in cylindrical geometry. It is also proved that trochoidal waves in cylindrical and plane geometry exhaust the class of nonlinear isobaric motions. Here and below by a wave in plane geometry we mean a wave in a uniform gravitational field which is characterized by the wave vector k. It is obvious that waves in both plane and cylindrical geometry are two-dimensional motions, since the fluid particles in motion are fixed in the plane and the motions in parallel planes are the same. The trochoidal wave in cylindrical geometry is of interest, since it describes a nonlinear wave on the surface of a cavity in a rotating fluid, a situation which is frequently encountered in applications.
KeywordsWave Vector Gravitational Field Nonlinear Wave Parallel Plane Hydrodynamic Equation
Unable to display preview. Download preview PDF.
- 1.J. F. von Gerstner, “Theorie der Wellen sammt einer abgeleiteten Theorie der Deichprofile,” Gilberts's Ann. Phys.,32, 412 (1809).Google Scholar
- 2.L. N. Sretenskii, The Theory of Wave Motions of Fluids [in Russian], Gostekhizdat, Moscow—Leningrad (1977).Google Scholar
- 3.H. Lamb, Hydrodynamics, Cambridge (1932).Google Scholar
- 4.L. M. Milne-Thomson, Theoretical Hydrodynamics, Macmillan, New York (1950).Google Scholar
- 5.M. L. Dubreil-Jacolin, “Sur les ondes de type permanent dans les liquides hétérogènes,” Rend. Accad. dei Lincei, Roma,15 (6), 314 (1932).Google Scholar
- 6.I. A. Kibel', “Certain plane motions in a heavy compressible fluid,” in Applied Mathematics and Mechanics, Vol. I [in Russian], Gostekhizdat, Leningrad (1933), pp. 51–55.Google Scholar
- 7.N. E. Kochin, Collected Works, Vol. 2 [in Russian], Izd. Akad. Nauk SSSH, Moscow-Leningrad (1949), pp. 80–85.Google Scholar
- 8.E. Ott, “Nonlinear evolution of the Rayleigh—Taylor instability of a thin layer,” Phys. Rev. Lett.,29, 1429 (1972).Google Scholar
- 9.Yu. A. Bashilov and S. V. Pokrovskii, “Taylor instability of a thin cylindrical layer,” Pis'ma Zh. Eksp. Teor. Fiz.,23, 462 (1976).Google Scholar
- 10.N. A. Inogamov, “Model analysis of Taylor instability of layers,” Pis'ma Zh. Tekh. Fiz.,3, 314 (1977).Google Scholar
- 11.G. N. Berman, The Cycloid [in Russian], Nauka, Moscow (1980).Google Scholar