Asymptotic analysis of a viscous fluid–thin plate interaction: Periodic flow

Grigory P. Panasenko; Ruxandra Stavre

doi:10.1016/j.crme.2012.06.001

Asymptotic analysis of a viscous fluid–thin plate interaction: Periodic flow

Grigory P. Panasenko ¹ ; Ruxandra Stavre ²

¹ Institute Camille-Jordan, UMR CNRS 5208, PRES University of Lyon, University of Saint-Etienne, 23, rue Dr Paul-Michelon, 42023 Saint-Etienne, France
² Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

Comptes Rendus. Mécanique, Volume 340 (2012) no. 8, pp. 590-595.

Résumé

The interaction “viscous fluid–thin plate” is considered when the thickness of the plate, ε, tends to zero, while the density and the Youngʼs modulus of the plate are of order $ε^{- 1}$ and $ε^{- 3}$ , respectively. The thickness of the fluid layer is of the order of one. An asymptotic expansion is constructed and the error estimates are proved. The leading term of the asymptotic expansion is the solution of the interaction problem “fluid-Kirchoff plate”. The method of asymptotic partial domain decomposition is discussed: the main part of the plate is described by a 1D model while a small part is simulated by the 2D elasticity equations, with appropriate junction conditions.

Reçu le : 2012-05-22
Accepté le : 2012-06-04
Publié le : 2012-07-19

DOI : 10.1016/j.crme.2012.06.001

Mots-clés : Fluid mechanics, Dynamical systems, Fluid structure interaction, Asymptotic analysis

Affiliations des auteurs :

Grigory P. Panasenko ¹ ; Ruxandra Stavre ²

¹ Institute Camille-Jordan, UMR CNRS 5208, PRES University of Lyon, University of Saint-Etienne, 23, rue Dr Paul-Michelon, 42023 Saint-Etienne, France
² Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania

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Grigory P. Panasenko; Ruxandra Stavre. Asymptotic analysis of a viscous fluid–thin plate interaction: Periodic flow. Comptes Rendus. Mécanique, Volume 340 (2012) no. 8, pp. 590-595. doi : 10.1016/j.crme.2012.06.001. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2012.06.001/

Bibliographie
Cité par

[1] G.P. Panasenko; R. Stavre Asymptotic analysis of a periodic flow in a thin channel with visco-elastic wall, J. Math. Pures Appl., Volume 85 (2006), pp. 558-579

[2] G. Panasenko Multi-Scale Modelling for Structures and Composites, Springer, 2005

A. Ćurković Error estimates for an effective model for the interaction between a thin fluid film and an elastic plate, Rad Hrvatske Akademije Znanosti i Umjetnosti. Matematičke Znanosti, Volume 564 (2025), pp. 329-345 | DOI:10.21857/ygjwrc27zy | Zbl:7990165
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Mario Bukal; Boris Muha Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid-structure interaction problem, Nonlinearity, Volume 35 (2022) no. 8, pp. 4695-4726 | DOI:10.1088/1361-6544/ac7d89 | Zbl:1504.35295
Mario Bukal; Boris Muha Rigorous derivation of a linear sixth-order thin-film equation as a reduced model for thin fluid-thin structure interaction problems, Applied Mathematics and Optimization, Volume 84 (2021) no. 2, pp. 2245-2288 | DOI:10.1007/s00245-020-09709-9 | Zbl:1487.35314
Mario Bukal; Boris Muha A review on rigorous derivation of reduced models for fluid-structure interaction systems, Waves in flows. The 2018 Prague-sum workshop lectures, Prague, Czech Republic, August 27–31, 2018, Cham: Birkhäuser, 2021, pp. 203-237 | DOI:10.1007/978-3-030-68144-9_8 | Zbl:1478.76004
Grigory P. Panasenko; Ruxandra Stavre Viscous fluid-thin elastic plate interaction: asymptotic analysis with respect to the rigidity and density of the plate, Applied Mathematics and Optimization, Volume 81 (2020) no. 1, pp. 141-194 | DOI:10.1007/s00245-018-9480-2 | Zbl:1431.76101
Ruxandra Stavre Optimization of the blood pressure with the control in coefficients, Evolution Equations and Control Theory, Volume 9 (2020) no. 1, pp. 131-151 | DOI:10.3934/eect.2020019 | Zbl:1431.49002
G. P. Panasenko; R. Stavre Three dimensional asymptotic analysis of an axisymmetric flow in a thin tube with thin stiff elastic wall, Journal of Mathematical Fluid Mechanics, Volume 22 (2020) no. 2, p. 35 (Id/No 20) | DOI:10.1007/s00021-020-0484-8 | Zbl:1434.74052
Grigory P. Panasenko; R. Stavre Viscous fluid-thin cylindrical elastic body interaction: asymptotic analysis on contrasting properties, Applicable Analysis, Volume 98 (2019) no. 1-2, pp. 162-216 | DOI:10.1080/00036811.2018.1442000 | Zbl:1409.74013
Matko Ljulj; Josip Tambača 3D structure-2D plate interaction model, Mathematics and Mechanics of Solids, Volume 24 (2019) no. 10, pp. 3354-3377 | DOI:10.1177/1081286519846202 | Zbl:7273370
Grigory P. Panasenko; A. E. Elbert Asymptotic of the solution of the contact problem for a thin elastic plate and a viscoelastic layer, Doklady Mathematics, Volume 97 (2018) no. 2, pp. 109-112 | DOI:10.1134/s1064562418020023 | Zbl:1452.74085
Frédéric Chardard; Alexander Elbert; Grigory Panasenko Asymptotic analysis of a thin elastic plate-viscoelastic layer interaction, Multiscale Modeling Simulation, Volume 16 (2018) no. 3, pp. 1258-1282 | DOI:10.1137/17m1138662 | Zbl:1411.35022
Rodolfo Venegas; Claude Boutin Acoustics of permeo-elastic materials, Journal of Fluid Mechanics, Volume 828 (2017), p. 135 | DOI:10.1017/jfm.2017.505
Irina Malakhova-Ziablova; Grigory Panasenko; Ruxandra Stavre Asymptotic analysis of a thin rigid stratified elastic plate – viscous fluid interaction problem, Applicable Analysis, Volume 95 (2016) no. 7, pp. 1467-1506 | DOI:10.1080/00036811.2015.1132311 | Zbl:1357.35087
Ruxandra Stavre A boundary control problem for the blood flow in venous insufficiency. The general case, Nonlinear Analysis. Real World Applications, Volume 29 (2016), pp. 98-116 | DOI:10.1016/j.nonrwa.2015.11.003 | Zbl:1331.49005
G. Panasenko; M.-C. Viallon Finite volume implementation of the method of asymptotic partial domain decomposition for the heat equation on a thin structure, Russian Journal of Mathematical Physics, Volume 22 (2015) no. 2, pp. 237-263 | DOI:10.1134/s1061920815020107 | Zbl:1327.35129
G. P. Panasenko; R. Stavre Viscous fluid-thin cylindrical elastic layer interaction: asymptotic analysis, Applicable Analysis, Volume 93 (2014) no. 10, pp. 2032-2056 | DOI:10.1080/00036811.2014.911843 | Zbl:1296.76125

Cité par 17 documents. Sources : Crossref, zbMATH

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