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## Archive of Issues

Russia Simferopol
Year
2021
Volume
31
Issue
2
Pages
311-330
 Section Mathematics Title The problem of normal oscillations of a viscous stratified fluid with an elastic membrane Author(-s) Tsvetkov D.O.a Affiliations Crimea Federal Universitya Abstract Normal oscillations of a viscous stratified fluid partially filling an arbitrary vessel and bounded above by an elastic horizontal membrane are studied. In this case, we consider a scalar model problem that reflects the main features of the vector spatial problem. The characteristic equation for the eigenvalues of the model problem is obtained, the structure of the spectrum and the asymptotics of the branches of the eigenvalues are studied. Assumptions are made about the structure of the oscillation spectrum of a viscous stratified fluid bounded by an elastic membrane for an arbitrary vessel. It is proved that the spectrum of the problem is discrete, located in the right complex half-plane symmetrically with respect to the real axis, and has a single limit point $+\infty$. Moreover, the spectrum is localized in a certain way in the right half-plane, the location zone depends on the dynamic viscosity of the fluid. Keywords stratification effect in viscous fluids, differential equation in Hilbert space, membrane, normal oscillations UDC 517.98 MSC 76D50, 34G10, 35P05 DOI 10.35634/vm210211 Received 19 March 2021 Language Russian Citation Tsvetkov D.O. The problem of normal oscillations of a viscous stratified fluid with an elastic membrane, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 311-330. References Lu D., Takizawa A., Kondo S. Overflow-induced vibration of a weir coupled with sloshing in a downstream tank, Journal of Fluids and Structures, 1997, vol. 11, issue 4, pp. 367--393. https://doi.org/10.1006/jfls.1997.0084 Lakis A.A., Paidoussis M.P. Free vibration of cylindrical shells partially filled with liquid, Journal of Sound and Vibration, 1971, vol. 19, issue 1, pp. 1-15. https://doi.org/10.1016/0022-460X(71)90417-2 Balendra T., Ang K.K., Paramasivam P., Lee S.L. Free vibration analysis of cylindrical liquid storage tanks, International Journal of Mechanical Sciences, 1982, vol. 24, issue 1, pp. 47-59. https://doi.org/10.1016/0020-7403(82)90020-0 Cheung Y.K., Zhou D. Coupled vibratory characteristics of a rectangular container bottom plate, Journal of Fluids and Structures, 2000, vol. 14, issue 3, pp. 339-357. https://doi.org/10.1006/jfls.1999.0272 Cheung Y.K., Zhou D. Hydroelastic vibration of a circular container bottom plate using the Galerkin method, Journal of Fluids and Structures, 2002, vol. 16, issue 4, pp. 561-580. https://doi.org/10.1006/jfls.2001.0430 Bauer H.F., Eidel W. Hydroelastic vibrations in a two-dimensional rectangular container filled with frictionless liquid and a partly elastically covered free surface, Journal of Fluids and Structures, 2004, vol. 19, issue 2, pp. 209-220. https://doi.org/10.1016/j.jfluidstructs.2003.11.002 Kopachevsky N.D., Krein S.G. Operator approach to linear problems of hydrodynamics. Vol. 1: self-adjoint problems for an ideal fluid, Basel: Birkhäuser, 2001. https://doi.org/10.1007/978-3-0348-8342-9 Kopachevsky N.D., Krein S.G. Operator approach to linear problems of hydrodynamics. Vol. 2: not self-adjoint problems for a viscous fluid, Basel: Birkhäuser, 2003. https://doi.org/10.1007/978-3-0348-8063-3 Zhang Y., Wen J., Xiao Y., Wen X., Wang J. Theoretical investigation of the sound attenuation of membrane-type acoustic metamaterials, Physics Letters A, 2012, vol. 376, issue 17, pp. 1489-1494. https://doi.org/10.1016/j.physleta.2012.03.010 Tariverdilo S., Mirzapour J., Shahmardani M., Rezazadeh G. Free vibration of membrane/bounded incompressible fluid, Applied Mathematics and Mechanics, 2012, vol. 33, issue 9, pp. 1167-1178. https://doi.org/10.1007/s10483-012-1613-8 Tarazaga P.A., Johnson M.E., Inman D.J. Experimental validation of the vibro-acoustic model of a pressurized membrane, Mechanical Systems and Signal Processing, 2014, vol. 45, issue 2, pp. 330-345. https://doi.org/10.1016/j.ymssp.2013.11.013 Eftekhari S.A. A differential quadrature procedure for free vibration of circular membranes backed by a cylindrical fluid-filled cavity, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2017, vol. 39, issue 4, pp. 1119-1137. https://doi.org/10.1007/s40430-016-0561-3 Kononov Yu.N., Dzhukha Yu.A. Influence of overload on axisymmetric oscillations of a circular membrane on the free surface of fluid in a cylindrical tank, Zbirnyk Prats' Instytutu Matematyky NAN Ukrainy, 2017, vol. 14, no. 2, pp. 32-41 (in Russian). https://zbmath.org/1399.76010 Amaouche M., Abderrahmane H.A. An exact eigenfrequency equation for the oscillations of a viscous fluid contained in an open and rectangular vessel with a flexible wall, European Journal of Mechanics - B/Fluids, 2018, vol. 70, pp. 1-5. https://doi.org/10.1016/j.euromechflu.2018.02.001 Kononov Yu.M., Shevchenko V.P., Dzhukh Yu.O. Axially symmetric vibrations of elastic annular bases and a perfect two-layer liquid in a rigid annular cylindrical vessel, Journal of Mathematical Sciences, 2019, vol. 240, issue 1, pp. 98-112. https://doi.org/10.1007/s10958-019-04338-2 Essaouini H., El Bakkali L., Capodanno P. Small oscillations of an ideal liquid contained in a vessel closed by an elastic circular plate, in uniform rotation, Theoretical and Applied Mechanics, 2017, vol. 44, issue 1, pp. 35-49. https://doi.org/10.2298/TAM160123002E Tsvetkov D.O. Oscillations of an ideal stratified fluid with an elastic membrane, Dinamicheskie Sistemy (Simferopol'), 2019, vol. 9 (37), no. 1, pp. 26-45 (in Russian). https://zbmath.org/1453.76031 Kononov Yu.M., Dzhukha Yu.O. Vibrations of two-layer ideal liquid in a rigid cylindrical vessel with elastic bases, Journal of Mathematical Sciences (New York), 2020, vol. 246, no. 3, pp. 365-383. https://zbmath.org/1448.74049 Choa I.H., Kimb M.H. Effect of a bottom-hinged, top-tensioned porous membrane baffle on the sloshing reduction in a rectangular tank, Applied Ocean Research, 2020, vol. 104, article number 102345. https://doi.org/10.1016/j.apor.2020.102345 Krejn M.G., Langer G.K. A contribution to the theory of quadratic pencils of self-adjoint operators, Soviet Mathematics. Doklady, 1964, vol. 5, pp. 266-269. https://zbmath.org/0198.47102 Krejn S.G. Oscillations of a viscous fluid in a container, Doklady Akademii Nauk SSSR, 1964, vol. 159, pp. 266-269 (in Russian). http://mi.mathnet.ru/eng/dan30338 Gabov S.A., Malysheva G.Yu. On a spectral problem connected with the oscillations of a viscous stratified fluid, USSR Computational Mathematics and Mathematical Physics, 1984, vol. 24, issue 3, pp. 170-174. https://doi.org/10.1016/0041-5553(84)90066-1 Essaouini H., El Bakkali L., Capodanno P. Mathematical study of the small oscillations of two nonmixing fluids, the lower inviscid, the upper viscoelastic, in an open container, Journal of Mathematical Fluid Mechanics, 2017, vol. 19, issue 4, pp. 645-657. https://doi.org/10.1007/s00021-016-0300-7 Zakora D.A. Spectral analysis of a viscoelasticity problem, Computational Mathematics and Mathematical Physics, 2018, vol. 58, no. 11, pp. 1761-1774. https://doi.org/10.1134/S0965542518110179 Tsvetkov D.O. Crumbled ice on the surface of a multilayered fluid, Siberian Electronic Mathematical Reports, 2020, vol. 17, pp. 777-801. https://doi.org/10.33048/semi.2020.17.056 Kopachevskii N.D., Orlova L.D., Pashkova Yu.S. Differential-operator and integro-differential equations in the problem of small oscillations of hydrodynamic systems, Uchenye Zapiski Simferopol'skogo Universiteta, 1995, vol. 41, no. 2, pp. 98-108 (in Russian). Tsvetkov D.O. On an initial-boundary value problem which arises in the dynamics of a viscous stratified fluid, Russian Mathematics, 2020, vol. 64, issue 8, pp. 50-63. https://doi.org/10.3103/S1066369X20080071 Kopachevsky N.D. Abstract Green formulas for triples of Hilbert spaces and sesquilinear forms, Journal of Mathematical Sciences, 2017, vol. 225, issue 2 pp. 226-264. https://doi.org/10.1007/s10958-017-3470-9 Kopachevskii N.D., Myshkis A.D. Hydrodynamics in weak fields of force. The small oscillations of a viscous liquid in a potential field of force, USSR Computational Mathematics and Mathematical Physics, 1966, vol. 6, no. 6, pp. 150-161. https://doi.org/10.1016/0041-5553(66)90167-4 Gokhberg I.C., Krein M.G. Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gil'bertovom prostranstve (Introduction to the theory of linear non-self-adjoint operators in Hilbert space), Moscow: Nauka, 1965. Full text