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Viscous Fluid–Thin Elastic Plate Interaction: Asymptotic Analysis with Respect to the Rigidity and Density of the Plate

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Abstract

A two-dimensional time dependent model of an interaction between a thin elastic plate and a Newtonian viscous fluid described by the non-steady Stokes equations is considered. It depends on a small parameter \(\varepsilon \) that is the ratio of the thicknesses of the plate and the fluid layer. The Young’s modulus of the plate and its density may be great or small parameters equal to some powers (positive or negative) of \(\varepsilon \) while the density and the viscosity of the fluid are supposed to be of order one. An asymptotic expansion is constructed and justified for various magnitudes of the rigidity and density of the plate. The limit problems are studied in all these cases. They are Stokes equations with some special coupled or uncoupled boundary conditions modeling the interaction with the plate. The estimates of the difference between the exact solution and a truncated asymptotic expansion are established. These estimates justify the asymptotic approximations.

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Acknowledgements

G. P. Panasenko was supported by the Russian Science Foundation operated by the Moscow Power Engineering Institute (Technical University) (Project 14-11-00306). This work was done during the stay of the R. Stavre at the Institute Camille Jordan as a visiting professor, supported by LABEX MILYON (ANR-10-LABX-0070) of University of Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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Authors and Affiliations

  1. Institute Camille Jordan UMR CNRS 5208 University of Lyon, University of Saint-Etienne, 23, rue Dr Paul Michelon, 42023, Saint-Étienne, France

    G. P. Panasenko

  2. Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 6, P.O. Box 1-764, 014700, Bucharest, Romania

    R. Stavre

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  1. G. P. Panasenko
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  2. R. Stavre
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Correspondence to G. P. Panasenko.

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Article dedicated to the 150th anniversary of the Romanian Academy.

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Panasenko, G.P., Stavre, R. Viscous Fluid–Thin Elastic Plate Interaction: Asymptotic Analysis with Respect to the Rigidity and Density of the Plate. Appl Math Optim 81, 141–194 (2020). https://doi.org/10.1007/s00245-018-9480-2

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