Application of homogenization theory related to Stokes flow in porous media.(English)Zbl 1059.76533

Stokes flow in porous media is considered. Known theoretical results are applied to obtain some homogenization results for the flow in question. The core of the paper is in numerical treatment of homogenization which is known to be practically impossible to attack directly for small mesh values. To this purpose a weak approximation of the original problem is constructed which has desired stability properties. Several examples of usefulness of the presented results are given.

MSC:

 76S05 Flows in porous media; filtration; seepage 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 49J45 Methods involving semicontinuity and convergence; relaxation 76D07 Stokes and related (Oseen, etc.) flows 76M25 Other numerical methods (fluid mechanics) (MSC2010) 76M50 Homogenization applied to problems in fluid mechanics

Keywords:

porous media; Stokes flow; homogenization
Full Text:

References:

 [1] Allaire, G.: Homogenization of the Stokes flow in connected porous medium. Asymptotic Analysis 3 (1989), 203-222. · Zbl 0682.76077 [2] Allaire, G.: Homogenization of the unsteady Stokes equation in porous media. Progress in Partial Differetial Equations: Calculus of variation, applications, Pitman Research notes in mathematics Series 267, New York, Longman Higher Education, 1992. · Zbl 0801.35103 [3] Arbogast, T., Douglas, J., Hornung, U.: Derivation of the double porosity modell of single phase flow via homogenization theory. SIAM J. Math. Anal. 21 (1990), 823-836. · Zbl 0698.76106 · doi:10.1137/0521046 [4] Bakhvalov, N., Panasenko, G.: Homogenization: Average Processes in Periodic Media. Dordrecht, Kluwer Academic Publishers, 1989. · Zbl 0692.73012 [5] Beliaev, A., Kozlov, S: Darcy equation for random porous media. Comm. Pure and Appl. Math. XLIX (1996), 1-34. DOI 10.1002/(SICI)1097-0312(199601)49:13.0.CO;2-J | [6] Bourgear, A., Carasso, C., Luckhaus, S., Mikelic, A.: Mathematical Modelling of Flow Through Porous Media. London: World Scientific, 1996. [7] Bourgeat, A., Mikelic, A.: Homogenization of a polymer flow through a porous medium. Nonlinear Anal., Theory Methods Appl. 26 (1996), 1221-1253. · Zbl 0863.76082 · doi:10.1016/0362-546X(94)00285-P [8] Donato, P., Saint Jean Paulin, J.: Stokes flow in a porous medium with a double periodicity. Progress in Partial Differetial Equations: the Metz Surveys, Pitman, Longman Press, 1994, pp. 116-129. · Zbl 0836.35116 [9] Ene, I., Saint Jean Paulin, J.: On a model of fractured porous media., 1996. · Zbl 0870.76084 [10] Ene, H., Poliševski, D.: Thermal Flow in Porous Media. Dordrecht, D. Reidel Publishing Company, 1987. [11] Holmbom, A.: Some Modes of Convergence and their Application to Homogenization and Optimal Composites Design. Doctoral Thesis, Luleå University of Technology, Sweden, 1996. [12] Lipton, R., Avellaneda. M.: A Darcy’s law for slow viscous flow through a stationary array of bubbles. Proc. Roy. Soc. Edinburgh 114A (1990,), 71-79. · Zbl 0850.76778 · doi:10.1017/S0308210500024276 [13] Lukkassen, D.: Some sharp estimates connected to the homogenized $$p$$-Laplacian equation. ZAMM-Z. angew. Math. Mech. 76 (1996), no. S2, 603-604. · Zbl 1126.35303 [14] Lukkassen, D.: Upper and lower bounds for averaging coefficients. Russian Math. Surveys 49 (1994), no. 4, 114-115. [15] Lukkassen, D.: On estimates of the effective energy for the Poisson equation with a $$p$$-Laplacian. Russian Math. Surveys 51 (1996), no. 4, 739-740. · Zbl 0871.35035 · doi:10.1070/RM1996v051n04ABEH002981 [16] Lukkassen, D.: Formulae and Bounds Connected to Homogenization and Optimal Design of Partial Differential Operators and Integral Functionals. Ph.D. thesis (ISBN: 82-90487-87-8), University of Tromsø, 1996. [17] Lukkassen, D., Persson, L.E., Wall, P.: On some sharp bounds for the homogenized $$p$$-Poisson equation (communicated by O.A. Oleinik). Applicable Anal. 58 (1995), 123-135. · Zbl 0832.35009 · doi:10.1080/00036819508840366 [18] Lukkassen, D., Persson, L.E., Wall, P.: Some engineering and mathematical aspects on the homogenization method. Composites Engineering 5 (1995), no. 5, 519-531. · doi:10.1016/0961-9526(95)00025-I [19] Meidell, A., Wall, P.: Homogenization and design of structures with optimal macroscopic behaviour. Proceedings of the 5’th International Conference on Computer Aided Optimum Design of Structures. To appear, 1997. [20] Mikelic, A.: Homogenization of the nonstationary Navier-Stoke equation in a domain with a grained boundary. Annali Mat. Pura. Appl. 158 (1991), 167-179. · Zbl 0758.35007 · doi:10.1007/BF01759303 [21] Milton, G. W.: On characterizing the set of possible effective tensors of composites: The Variational Method and the Translation Method. Comm. on Pure and Appl. Math. XLIII (1990), 63-125. · Zbl 0751.73041 · doi:10.1002/cpa.3160430104 [22] Nandakumaran, A. K.: Steady and evolution Stokes Equation in porous media with non-homogeneous boundary data; a homogenization progress. Differential and Integral equations 5 (1992), 73-93. · Zbl 0761.35007 [23] Persson, L. E., Persson, L., Svanstedt, N., Wyller, J.: The Homogenization Method: An Introduction. Lund, Studentlitteratur, 1993. · Zbl 0847.73003 [24] Sanches-Palencia, E.: Non Homogeneous Media and Vibration Theory. Lecture Notes in Physics 127. New York, Springer Verlag, 1980. [25] Tartar, L.: Incompressible fluid flow through porous media-Convergence of the homogenization process. Appendix of, 1980.
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