Abstract
We study the problem on small motions and normal oscillations of a system of two heavy immiscible stratified fluids partially filling a fixed vessel. The lower fluid is assumed to be viscous, while the upper one is assumed to be ideal. We find sufficient existence conditions for a strong (with respect to the time variable) solution of the initial-boundary value problem describing the evolution of the specified hydraulic system. For the corresponding spectral system, we obtain results about the localization of the spectrum, asymptotic behavior of branches of eigenvalues, and existence of the substantial spectrum of the problem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Avakyan, “Asymptotic distribution of a spectrum of a linear pencil perturbed by an analytic operator-function,” Funktsional. Anal. i Prilozhen., 12, No. 2, 66-67 (1978).
T.Ya. Azizov, V. Hardt, N. D. Kopachevskiĭ, and R. Mennicken, “To the problem on small motions and normal oscillations of a viscous fluid in a partially filled container,” Math. Nachr., 248-249, 3–39 (2003).
A. E. Bukatov and L. V. Cherkesov, Waves in Inhomogeneous Sea [in Russian], Naukova Dumka, Kiev (1983).
L. V. Cherkesov, Hydrodynamics of Surface and Internal Waves [in Russian], Naukova Dumka, Kiev (1976).
S. A. Gabov and A. G. Sveshnikov, Problems of the Dynamics of Stratified Fluids [in Russian], Nauka, Moscow (1986).
S. A. Gabov and A. G. Sveshnikov, Linear Problems in the Theory of Nonstationary Internal Waves [in Russian], Nauka, Moscow (1990).
E. Gagliargo, “Caratterizzazioni delle trace sulla frontiera relative ad alcune classi di funzioni in n variabili,” Rend. Sem. Mat. Univ. Padova, 27, 284–305 (1957).
N. D. Kopachevskiĭ, S. G. Kreĭn, and Ngo Zui Kan, Operator Methods in Linear Hydrodynamics: Evolution and Spectral Problems [in Russian], Nauka, Moscow (1989).
N. D. Kopachevskiĭ and A. N. Temnov, “Free oscillations of a viscous stratified fluid in a vessel,” Deposited in VINITI, No. 4531-83 (1985).
N. D. Kopachevskiĭ and A. N. Temnov, “Oscillations of a stratified fluid in a reservoir of arbitrary shape,” Zh. Vychisl. Mat. Mat. Fiz., 26, No. 5, 734–755 (1986).
N. D. Kopachevskiĭ and A. N. Temnov, “Oscillations of an ideal stratified fluid in a cylindrical reservoir for a variable floatability frequency,” Differ. Uravn., 24, No. 10, 1784–1796 (1988).
V. K. Krauss, Internal Waves [in Russian], Gidrometeoizdat, Leningrad (1968).
S. G. Kreĭn, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1967).
A. S. Markus, Introduction into the Spectral Theory of Polynomial Operator Pencils [in Russian], Shtiintsa, Kishinev (1986).
Yu. Z. Miropol’skĭ, Ocean Dynamics of Internal Gravitational Waves [in Russian], Gidrometeoizdat, Leningrad (1981).
M. B. Orazov, Certain Issues of Spectral Theory of Non-Self-Adjoint Operators and Related Problems of Mechanics, D. Sc. thesis, Ashkhabad (1982).
T.P. Temchenko, Spectral and Evolution Oscillation Problems for Multi-Layer Stratified Fluids, Ph.D. thesis, Simferopol’ (1989).
D. O. Tsvetkov, “Small motions of a stratified fluid in a rotating vessel,” Tavrich. Vestn. Mat. Inf., No. 1, 140–149 (2003).
D. O. Tsvetkov, “On normal oscillations of a partially dissipative hydraulic system,” Uch. Zap. Tavrich. Nat. Univ. Mat., Mech., Inf., Kib., 16 (55), No. 2, 113–122 (2003).
J. Turner, Buoyancy Effects in Fluids [Russian translation], Mir, Moscow (1977).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 29, Proceedings of KROMSH, 2008.
Rights and permissions
About this article
Cite this article
Kopachevskii, N.D., Tsvetkov, D.O. Oscillations of stratified fluids. J Math Sci 164, 574–602 (2010). https://doi.org/10.1007/s10958-010-9764-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-9764-9