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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 152743, 47 pages
http://dx.doi.org/10.1155/2012/152743
Research Article

A Viscous Fluid Flow through a Thin Channel with Mixed Rigid-Elastic Boundary: Variational and Asymptotic Analysis

1Institut Camille Jordan UMR, CNRS 5208, PRES University of Lyon/University of Saint-Etienne, 23 Rue Paul Michelon, 42023 Saint-Etienne, France
2Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014 700 Bucharest, Romania

Received 23 January 2012; Accepted 29 February 2012

Academic Editor: D. O'Regan

Copyright © 2012 R. Fares et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the nonsteady Stokes flow in a thin tube structure composed by two thin rectangles with lateral elastic boundaries which are connected by a domain with rigid boundaries. After a variational approach of the problem which gives us existence, uniqueness, regularity results, and some a priori estimates, we construct an asymptotic solution. The existence of a junction region between the two rectangles imposes to consider, as part of the asymptotic solution, some boundary layer correctors that correspond to this region. We present and solve the problems for all the terms of the asymptotic expansion. For two different cases, we describe the order of steps of the algorithm of solving the problem and we construct the main term of the asymptotic expansion. By means of the a priori estimates, we justify our asymptotic construction, by obtaining a small error between the exact and the asymptotic solutions.