Analysis of the Oscillations of Stratified Liquid with Elastic Ice
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Keywords

Strong solution
Cauchy problem
Spectral problem
Initial-boundary value problem
Differential equation in Hilbert space

How to Cite

Tsvetkov, D. . (2022). Analysis of the Oscillations of Stratified Liquid with Elastic Ice. Journal of Advances in Applied & Computational Mathematics, 8, 87–97. https://doi.org/10.15377/2409-5761.2021.08.6

Abstract

We study the problem of small motions of an ideal stratified liquid with a free surface totally covered by an elastic ice. The elastic ice is modeled by an elastic plate. We reduce the original initial boundary value problem to an equivalent Cauchy problem for a second-order differential equation in a Hilbert space. We obtain conditions under which there exists a strong (with respect to time) solution of the initial boundary value problem describing the evolution of the hydrodynamic system under consideration. We also study the spectrum of normal oscillations, the basic properties of the eigenfunctions.

https://doi.org/10.15377/2409-5761.2021.08.6
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References

Shaik VA, Ardekani AM. Squirming in density-stratified fluids. Phys Fluids 2021; 33: (101903). https://doi.org/10.1063/5.0065958

Akiyama S, Waki Y, Okino S, Hanazaki H. Unstable jets generated by a sphere descending in a very strongly stratified fluid. J Fluid Mech. 2019; 867: 26-44. https://doi.org/10.1017/jfm.2019.123

Zhang J, Mercier MJ, Magnaudet J. Core mechanisms of drag enhancement on bodies settling in a stratified fluid. J Fluid Mech 2019; 875: 622-656. https://doi.org/10.1017/jfm.2019.524

More RV, Ardekani FM. Hydrodynamic interactions between swimming microorganisms in a linearly density stratified fluid. Phys Rev E 2021; 103: (013109). https://doi.org/10.1103/PhysRevE.103.013109

Camassa R, Falqui G, Ortenzi G. Hamiltonian aspects of Tthree-layer stratified Ffluids. J Nonlinear Sci. 2021; 31: (70). https://doi.org/10.1007/s00332-021-09726-0

Sreenivasan B, Maurya G. Evolution of forced magnetohydrodynamic waves in a stratified fluid. J Fluid Mech. 2021; 922: (A32). https://doi.org/10.1017/jfm.2021.565

Samodurov AS, Chukharev AM, Kazakov DA. Basic regularities of vertical turbulent exchange in the mixed and stratified layers of the black sea. Phys Oceanogr. 2021; 28(4): 376-391. https://doi.org/10.22449/1573-160X-2021-4-376-391

Zakora DA. Spectral analysis of a viscoelasticity problem. Comput Math Mathemat Phys 2018; 58(11): 1761-1774. https://doi.org/10.1134/S0965542518110179

Forduk KV, Zakora DA. Problem on small motions of a system of bodies filled with ideal fluids under the action of an elastic damping device. Lobachevskii J Math. 2021; 42(5): 889-900. https://doi.org/10.1134/S199508022105005X

Forduk KV, Zakora DA. A problem of normal oscillations of a system of bodies partially filled with ideal fluids under the action of an elastic damping device. Sib Electron Math Rep. 2021; 18(2): 997-1014. https://doi.org/10.33048/semi.2021.18.075

Tsvetkov DO. Oscillations of a liquid partially covered with ice. Lobachevskii J Math. 2021; 42(5): 1078-1093. https://doi.org/10.1134/S199508022105019X

Tsvetkov DO. Oscillations of a stratified liquid partially covered with ice (general case). Math Notes 2020; 107(1): 160-172. https://doi.org/10.1134/S0001434620010150

Tsvetkov DO. Small motions of an ideal stratified fluid with a free surface completely covered with the elastic ice. Sib Elektron Math Izv. 2018; 15: 422-435.

Essaouini H, Bakkali L, Capodanno P. Mathematical study of the small oscillations of two nonmixing fluids, the lower inviscid, the upper viscoelastic, in an open container. J Math Fluid Mech 2017; 19: 645-657. https://doi.org/10.1007/s00021-016-0300-7

Tsvetkov DO. On an initial-boundary value problem which arises in the dynamics of a viscous stratified fluid. Russ Math. 2020; 64: 50-63. https://doi.org/10.3103/S1066369X20080071

Tsvetkov DO. The problem of normal oscillations of a viscous stratified fluid with an elastic membrane. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Kompyuternye Nauki 2021; 31(2): 311-330. https://doi.org/10.35634/vm210211

Kopachevsky ND, Krein SG. Operator approach to linear problems of hydrodynamics. Vol. 1. Self-adjoint problems for an ideal fluid. Basel, Boston, Berlin, Birkhauser 2001. https://doi.org/10.1007/978-3-0348-8342-9_1

Voytitsky VI. Strong dissipative hydrodynamical systems and the operator pencil of S. Krein. Lobachevskii J Math. 2021; 42(5): 1094-1112. https://doi.org/10.1134/S1995080221050206

Goldstein JA. Semigroups of linear operators and applications. Oxford University Press 1985.

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