Abstract
We consider a sequence of the Dirichlet problems for steady Navier- Stokes equations in domains perforated with channels where are closed subsets of bounded domain containedin small neighborhoods of some lines. While the number I (s) of channels tends toinfinity as these small sets are thinned.We study the asymptotic behaviorof solutions us (x) to problems in domains with thin channels. Wefind conditions on perforated domains under which sequence of solutions converges to solution of homogenized problem as. The proof is based on the asymptotic expansion of us (x) and on pointwise and integral estimates of auxiliary functions which are solutions of model boundary value problems.
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References
Allaire, G.: Homogenization of the Stokes flow in a connected porous medium. Asymptot. Anal. 2, 203–222 (1989)
Allaire, G.: One-phase newtonian flow. In Homogenization and Porous Media, ed. U.Hornung, pp. 45–68. Springer, New York (1997)
Beliaev, A.Yu, Kozlov, S.M.: Darcy equation for random porous media. Comm. Pure Appl. Math. 49, 1–34 (1995)
Berezhnyi, M., Berlyand L., Khruslov E.: Homogenized models of complex fluids, Networks and heterogeneous media. 3 (4), 831–862 (2008)
Bourgeaut, A., Mikelic, A.: Note on the Homogenization of Bingham Flow through Porous Medium. J. Math. Pures Appl. 72, 405–414 (1993)
Byhovskij, E.B., Smirnov, N.V.: Orthogonal decomposition of the space of vector functions square-summable on a given domain, and the operators of vector analysis. Works of Steklov Institute of Mathematics AN SSSR, M.-L. 59, 5–35 (1960)
Clopeau, Th., Ferrin, Gilbert, R. P., Mikelic, A.: Homogenizing the acoustic properties of the Seabed, II, Math. Comp. Model. 32, 821–841, (2001)
Ene, H.I., Sanchez-Palencia, E.: Equations et phénomenènes de surface pour l’écoulement dans un modèle de milieu poreux. J. Mécan. 14, 73–108 (1975)
Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume I, Linearized steady problems, Springer Tracts in Nat. Philos., Vol. 38 (1994)
Gilbert, R.P., Mikelic, A.: Homogenizing the acoustic properties of the seabed, Part I. Nonlinear Anal. 40, 185–212 (2000)
Jäger, W., Mikelic, A.: On the effective equations for a viscous incompressible fluid flow through a filter of finite thickness. Commun. Pure Appl. Math. LI, 1073–1121 (1998)
Jikov, V.V., Kozlov, S., Oleinik, O.: Homogenization of Differential Operators and Integral Functionals. Springer Verlag, New York (1994)
Khruslov, E.Ya., Lvov, V.A.: Boundary problems for the linearized Navier-Stokes system in domains with a fine-grained boundary (Russian). Vestn. Khar’kov. Univ. 119, Mat. Mekh. 40, 3–22 (1975)
Khruslov, E.Ya., Lvov, V.A.: Boundary problems for the general Navier-Stokes system in domains with a fine-grained boundary (Russian). Vestn. Khar’kov. Univ. 119. Mat. Mekh. 40, 23–36 (1975)
Kondratiev, V.A., Oleinik, O.A.: On Korn’s inequalities. C.R. Acas. Sci. Paris 308, Serie I, 483–487 (1989)
Ladyzhenskaya, O.A.: Mathematical Problems of the Dynamics of Viscous Incompressible Fluids. Nauka, Moscow (1970)
Marchenko, V.A., Khruslov, E.Ya.: Kraevye zadaci v oblastjah s melkozernistoi granicei (Russian). Kiev, “Naukova Dumka” (1974)
Marchenko, V.A., Khruslov, E.Ya.: Homogenization of Partial Differential Equations. Series: Prog. Math. Phys., XII, Vol. 46 (2006)
Namlyeyeva, Yu., Nečasová, Š.: Homogenization of the steady Navier-Stokes equations in domains with a fine-grained boundary. Proceedings of the Conference ”Topical problems of the fluid mechanics 2006. Institute of Thermomechanics AS CR 2006, 103–106 (2006)
Nitsche, J.A.: On Korn’s second inequality. R.A.I.R.O., Analyse Numerique 15(3) 237–248 (1981)
Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics, 127, Springer, IX (1980)
Skrypnik, I.V.: Methods for Analysis of Nonlinear Elliptic Boundary Value Problems. American Mathematical Society, Providence, RI (1994)
Skrypnik, I.V.: Averaging of quasilinear parabolic problems in domains with fine-grained boundaries. Diff. Equ. 31(2), 327–339 (1995)
Tartar, L.: Navier- Stokes Equations. Elsevier Science Publishers B. V., Amsterdam (1984)
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Namlyeyeva, Y.V., Nečcasová, Š., Skrypnik, I. (2010). The Dirichlet Problems for Steady Navier-Stokes Equations in Domains with Thin Channels. In: Rannacher, R., Sequeira, A. (eds) Advances in Mathematical Fluid Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04068-9_22
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DOI: https://doi.org/10.1007/978-3-642-04068-9_22
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