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On homogenization of stokes flow in slowly varying media with applications to fluid–structure interaction

  • Donald L. BrownEmail author
  • Peter Popov
  • Yalchin Efendiev
Original Paper

Abstract

In this paper we establish corrector estimates for Stokes flow in slowly varying perforated media via two scale asymptotic analysis. Current methods and techniques are often not able to deal with changing geometries prevalent in applied problems. For example, in a deformable porous medium environment, the geometry does not remain periodic under mechanical deformation and if slow variation in the geometry occurs. For such problems, one cannot use classical homogenization results directly and new homogenization results and estimates are needed. Our work uses asymptotic techniques of Marusic-Paloka and Mikelic (Bollettino U.M.I 7:661–671, 1996) where the authors constructed a downscaled velocity which converges to the fine-scale velocity at a rate of ε 1/6 where ε is the characteristic length scale. We assume a slowly varying porous medium and study homogenization and corrector estimates for the Stokes equations. Slowly varying media arise, e.g., in fluid–structure interaction (FSI) problems (Popov et al. in Iterative upscaling of flows in deformable porous media, 2008), carbonation of porous concrete (Peter in C. R. Mecanique 335:357–362, 2007a; C. R. Mecanique 335:679–684, 2007b), and various other multiphysics processes. To homogenize Stokes flows in such media we restate the cell problems of Marusic-Paloka and Mikelic (Bollettino U.M.I 7:661–671, 1996) in a moving RVE framework. Further, to recover the same convergence properties it is necessary to solve an additional cell problem and add one more corrector term to the downscaled velocity. We further extend the framework of Marusic-Paloka and Mikelic (Bollettino U.M.I 7:661–671, 1996) to three spatial dimensions in both periodic and variable pore-space cases. Next, we also propose an efficient algorithm for computing the correctors by solving a limited number of cell problems at selected spatial locations. We present two computational examples: one for a constructed medium of elliptical perforations, and another for a fractured medium with FSI driven deformation. We obtain numerical estimates that confirm the theory in these two examples.

Keywords

Stokes Homogenization Fluid–structure interaction Slowly varying geometry Multiscale 

Mathematics Subject Classification (2000)

35A25 74Q99 35B27 74F10 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Donald L. Brown
    • 1
    Email author
  • Peter Popov
    • 2
  • Yalchin Efendiev
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria

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