Homogeni z ation and porous media

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

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

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

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

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

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

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

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 homogenizationof the diffusion problem is given by where is given by

11. Proposition 2: • The tensor A is symmetric • 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: • If , then • The coefficients are given by

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

16. e Media with obstacles • Medium has periodic arrangement of obstacles

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

18. Diffusion problem • Diffusion only in • Assumptions: and

19. Substitution of expansion • Comparing terms of different powers yields

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

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

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 homogenizationof 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 sameorder • 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 tensorKis 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