1. Homogenization and porous media Heike Gramberg, CASA Seminar Wednesday 23 February 2005 by Ulrich Hornung Chapter 1: Introduction

2. Diffusion in periodic media –Special case: layered media Diffusion in media with obstacles Stokes problem: derivation of Darcy’s law Overview

3. We start with the following Problem Let with bounded and smooth Diffusion equation Diffusion equation (Review)

4. is rapidly oscillating i.e. a=a(y) is Y-periodic in with periodicity cell Assumptions

5. has an asymptotic expansion of the form and are treated as independent variables Ansatz

6. Comparing terms of different powers yields where Substitution of expansion

7. Terms of order : since is Y-periodic we find Solution

8. Terms of order : separation of variables where is Y-periodic solution of

9. Terms of order integration over Y using for all Y-periodic g(y) :

10. Propositions Proposition 1: The homogenization of the diffusion problem is given by where is given by

11. Proposition 2: a.The tensor A is symmetric b.If a satisfies a(y)> a >0 for all y then A is positive definite

12. Remarks are uniquely defined up to a constant are uniquely defined Problem can be generalized by considering Eigenvalues l of A satisfy Voigt-Reiss inequality: where

13. Example: Layered Media Assumption: Then and is Y-periodic solution of

14. Proposition 3: a)If, then b)The coefficients are given by

15. Remarks Effective Diffusivity in direction parallel to layers is given by arithmetic mean of a(y) Effective Diffusivity in direction normal to layers is given by geometric mean of a(y) Extreme example of Voigt-Reiss inequality

16. Media with obstacles Medium has periodic arrangement of obstacles 

17. Standard periodicity cell Geometric structure within Assumption: Formal description of geometry

18. Diffusion problem Diffusion only in Assumptions: and

19. Comparing terms of different powers yields Substitution of expansion

20. Lemmas Lemma 1: for and Lemma 2 (Divergence Theorem): for Y-periodic

21. Terms of order : for using Lemmas 1 and 2 we find therefore Solution

22. Terms of order : for with boundary condition for

23. separation of variables where is Y-periodic solution of

24. Terms of order using Lemma 2 and boundary conditions: hence is solution of

25. Proposition Proposition 4: The homogenization of the diffusion problem on geometry with obstacles is given by where is given by

26. Remarks Due to the homogeneous Neumann conditions on integrals over boundary disappear Weak formulation of the cell problem where is characteristic function of

27. Stokes problem For media with obstacles Assumptions

28. Solution Comparing coefficients of the same order –Stokes equation: –Conservation of mass : –Boundary conditions:

29. With we get for Separation of variables for both where are solution of

30. Darcy’s law Averaging velocity over where is given by

31. Conservation of mass Term of order in conservation of mass Integration over yields

32. Proposition Proposition 5: The homogenization of the Stokes problem is given by Proposition 6: The tensor K is symmetric and positive definite

33. Conclusions We have looked at homogenization of the Diffusion problem and the Stokes problem on media with obstacles Solutions of the homogenized problems can be expressed in terms of solutions of cell problems The homogenization of the Stokes problem leads to the derivation of Darcy’s law