Abstract
The goal of this paper is to present a different approach to the homogenization of the Dirichlet boundary value problem in porous medium. Unlike the standard energy method or the method of two-scale convergence, this approach is not based on the weak formulation of the problem but on the very weak formulation. To illustrate the method and its advantages we treat the stationary, incompressible Navier-Stokes system with the non-homogeneous Dirichlet boundary condition in periodic porous medium. The nonzero velocity trace on the boundary of a solid inclusion yields a non-standard addition to the source term in the Darcy law. In addition, the homogenized model is not incompressible.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. Allaire: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Ration. Mech. Anal. 113 (1991), 209–259.
G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–1518.
N. S. Bakhvalov, G. Panasenko: Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials. Mathematics and Its Applications: Soviet Series 36. Kluwer Academic, Dordrecht, 1989.
A. Capatina, H. Ene: Homogenisation of the Stokes problem with a pure non-homogeneous slip boundary condition by the periodic unfolding method. Eur. J. Appl. Math. 22 (2011), 333–345.
D. Cioranescu, P. Donato, H. Ene: Homogenization of the Stokes problem with non-homogeneous slip boundary conditions. Math. Methods Appl. Sci. 19 (1996), 857–881.
D. Cioranescu, P. Donato, R. Zaki: The periodic unfolding method in perforated domains. Port. Math. (N.S.) 63 (2006), 467–496.
C. Conca: Étude d’un fluide traversant une paroi perforée. I. Comportement limite près de la paroi. J. Math. Pures Appl., IX. Sér. 66 (1987), 1–43. (In French.)
C. Conca: Étude d’ un fluide traversant une paroi perforée. II. Comportement limite loin de la paroi. J. Math. Pures Appl., IX. Sér. 66 (1987), 45–69. (In French.)
A. E. Craig, J. O. Dabiri, J. R. Koseff: A kinematic description of the key flow characteristics in an array of finite-height rotating cylinders. J. Fluids Eng. 138 (2016), Article ID 070906, 16 pages.
L. Diening, E. Feireisl, Y. Lu: The inverse of the divergence operator on perforated domains with applications to homogenization problems for the compressible Navier-Stokes system. ESAIM, Control Optim. Calc. Var. 23 (2017), 851–868.
R. Lipton, M. Avellaneda: Darcy’s law for slow viscous flow past a stationary array of bubbles. Proc. R. Soc. Edinb., Sect. A 114 (1990), 71–79.
E. Marušić-Paloka: Solvability of the Navier-Stokes system with L2 boundary data. Appl. Math. Optimization 41 (2000), 365–375.
E. Marušić-Paloka, A. Mikelić: The derivation of a nonlinear filtration law including the inertia effects via homogenization. Nonlinear Anal., Theory Methods Appl., Ser. A 42 (2000), 97–137.
A. Mikelić, I. Aganović: Homogenization in a porous media under a nonhomgeneous boundary condition. Boll. Unione Mat. Ital., VII. Ser., A 1 (1987), 171–180.
G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623.
E. Sanchez-Palencia: Non-Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127. Springer, Berlin, 1980.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author has been supported by the Croatian Science Foundation (grant: 2735 Asymptotic analysis of the boundary value problems in continuum mechanics — AsAn)
Rights and permissions
About this article
Cite this article
Marušić-Paloka, E. Application of very weak formulation on homogenization of boundary value problems in porous media. Czech Math J 71, 975–989 (2021). https://doi.org/10.21136/CMJ.2021.0161-20
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2021.0161-20