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

Handle URI:
http://hdl.handle.net/10754/599039
Title:
On homogenization of stokes flow in slowly varying media with applications to fluid–structure interaction
Authors:
Brown, Donald L.; Popov, Peter; Efendiev, Yalchin
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 ε1/6 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. © 2011 Springer-Verlag.
Citation:
Brown DL, Popov P, Efendiev Y (2011) On homogenization of stokes flow in slowly varying media with applications to fluid–structure interaction. Int J Geomath 2: 281–305. Available: http://dx.doi.org/10.1007/s13137-011-0025-y.
Publisher:
Springer Nature
Journal:
GEM - International Journal on Geomathematics
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
11-Sep-2011
DOI:
10.1007/s13137-011-0025-y
Type:
Article
ISSN:
1869-2672; 1869-2680
Sponsors:
This publication is based in part on work supported by Award No. KUS-C1-016-04made by King Abdullah University of Science and Technology (KAUST) and US National Science Foun-dation grant NSF-DMS-0811180. We would like to also acknowledge partial support from NSF (724704,0811180, 0934837) and the DOE.
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Full metadata record

DC FieldValue Language
dc.contributor.authorBrown, Donald L.en
dc.contributor.authorPopov, Peteren
dc.contributor.authorEfendiev, Yalchinen
dc.date.accessioned2016-02-25T13:51:41Zen
dc.date.available2016-02-25T13:51:41Zen
dc.date.issued2011-09-11en
dc.identifier.citationBrown DL, Popov P, Efendiev Y (2011) On homogenization of stokes flow in slowly varying media with applications to fluid–structure interaction. Int J Geomath 2: 281–305. Available: http://dx.doi.org/10.1007/s13137-011-0025-y.en
dc.identifier.issn1869-2672en
dc.identifier.issn1869-2680en
dc.identifier.doi10.1007/s13137-011-0025-yen
dc.identifier.urihttp://hdl.handle.net/10754/599039en
dc.description.abstractIn 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 ε1/6 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. © 2011 Springer-Verlag.en
dc.description.sponsorshipThis publication is based in part on work supported by Award No. KUS-C1-016-04made by King Abdullah University of Science and Technology (KAUST) and US National Science Foun-dation grant NSF-DMS-0811180. We would like to also acknowledge partial support from NSF (724704,0811180, 0934837) and the DOE.en
dc.publisherSpringer Natureen
dc.subjectFluid-structure interactionen
dc.subjectHomogenizationen
dc.subjectMultiscaleen
dc.subjectSlowly varying geometryen
dc.subjectStokesen
dc.titleOn homogenization of stokes flow in slowly varying media with applications to fluid–structure interactionen
dc.typeArticleen
dc.identifier.journalGEM - International Journal on Geomathematicsen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionBulgarian Academy of Sciences, Sofia, Bulgariaen
kaust.grant.numberKUS-C1-016-04en
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