Homogenization and porous media Heike Gramberg, CASA Seminar Wednesday 23 February 2005 by Ulrich Hornung Chapter 1: Introduction
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Homogenization and porous media
Heike Gramberg, CASA Seminar Wednesday 23 February 2005
by Ulrich Hornung
Chapter 1: Introduction
• Diffusion in periodic media– Special case: layered media
• Diffusion in media with obstacles
• Stokes problem: derivation of Darcy’s law
Overview
We start with the following Problem
• Let with bounded and smooth
• Diffusion equation
Diffusion equation (Review)
NR
0
D
a x u x f x x
u x u x x
• is rapidly oscillating i.e.
• a=a(y) is Y-periodic in with periodicity cell
Assumptions
a
xa x a x for all
NR
1 0 1 1N iY y y y y i N , , , , ,
• has an asymptotic expansion of the form
• and are treated as independent variables
Ansatz
u
20 1 2u x u x y u x y u x y , , ,
x xy
1x y
• Comparing terms of different powers yields
where
Substitution of expansion
20
11 0
02 1
0
0
0
0
,:
: , ,
: , ,
,
yy
yy xy yx
yy xy yx
xx
u x y
u x y u x y
u x y u x y
u x y f x
A
A A A
A A A
A
ab a ba y A :
• Terms of order :
since is Y-periodic we find
Solution2
0 0u x y u x,
0 ,u x y
0 0y ya y u x y ,
• Terms of order :
separation of variables
where is Y-periodic solution of
y y j y ja y w y a y e
1
1 01
j j
N
y y y xj
a y u x y a y u x
,
1 01
, j
N
j xj
u x y w y u x
jw y
• Terms of order integration over Y
using for all Y-periodic g(y):
0 :
2 1
0 0
, ,
yy xy yx
xx
u x y u x y
u x f x
A A A
A
Y
dy
0iyYg y dy
01
0,
i i j
N
y j ij x xYi j
a y w y dy u f x
PropositionsProposition 1: The homogenization of the
diffusion problem is given by
where is given by
0
D
A u x f x x
u x u x x
ijA a
iij y j ijY
a a y w y dy
Proposition 2:
a. The tensor A is symmetric
b. If a satisfies a(y)>>0 for all y then
A is positive definite
Remarks
• are uniquely defined up to a constant• are uniquely defined• Problem can be generalized by considering
• Eigenvalues of A satisfy Voigt-Reiss inequality:
where
jw
, a a x y A A x
ija
11a a
:Y
f f dy
Example: Layered Media
• Assumption: • Then
and is Y-periodic solution of
1, , N Na y y a y
1 0
0
for
for
N N
N N
d ddy dyN j N
d ddy dyN j N
a y w y j N
a y w y j N
1, ,j N j Nw y y w y
jw
Proposition 3:
a) If , then
b) The coefficients are given by
01 1
0
, ,
Ny da
N N Nd
a
w y y y
0 for andjw j N
ija11 for
otherwisea
ij
ij
i j Na
a
1, , N Na y y a y
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
Media with obstacles
• Medium has periodic arrangement of obstacles
B
G
• Standard periodicity cell
• Geometric structure within
• Assumption:
Formal description of geometry
B
G
\ B = B G = B
Diffusion problem
• Diffusion only in
• Assumptions: and
B
0
0
D
a x u x f x x
a x u x x
u x u x
B
xa x a
20 1 2u x u x y u x y u x y , , ,
• Comparing terms of different powers yieldsSubstitution of expansion
20
11 0
02 1
0
0
0
0
yy
yy xy yx
yy xy yx
xx
u x y
u x y u x y
u x y u x y
u x y f x
A
A A A
A A A
A
,:
: , ,
: , ,
,
10
00 1
11 2
0
0
0
,:
: , ,
: , ,
y
x y
x y
a y u x y
a y u x y u x y
a y u x y u x y
with boundary conditions on
Lemmas
• Lemma 1: for and
• Lemma 2 (Divergence Theorem):
for Y-periodic
,g g x y
y y yf g f g f g
,g g x y
, ,y g x y dy g x y d y
B
,f f x y
• Terms of order : for
using Lemmas 1 and 2 we find
therefore
Solution2
0 0u x y u x,
0 0y ya y u x y ,
yB
0 0
20
0 0
0 ,
| , |
, ,
y y
y
y
u a y u x y dy
a y u x y dy
u x y a y u x y d y
B
B
• Terms of order : for
• with boundary condition for
1
1 01
j j
N
y y y xj
a y u x y a y u x
,
yB
1u on
1 01
j
N
y j xj
u x y u x
,
• separation of variables
where is Y-periodic solution of
1 01
, j
N
j xj
u x y w y u x
jw y
y y j y j
y j j
a y w y a y e y
w x y e y
,
B
• Terms of order
using Lemma 2 and boundary conditions:
hence is solution of
0 :
2 1
0 0
, ,
yy xy yx
xx
u x y u x y
u x f x
A A A
A
Bdy
01
0,
i i j
N
y j ij x xi j
a y w y dy u f x
B
2 1 2 1
0
yy yx y xu u dy a y u u d y
BA A
0 0u u x
PropositionProposition 4: The homogenization of the
diffusion problem on geometry with obstacles is given by
where is given by
0
D
A u x f x x
u x u x x
ijA a
iij y j ija a y w y dy B
Remarks• Due to the homogeneous Neumann conditions
on integrals over boundary disappear• Weak formulation of the cell problem
where is characteristic function of
0,j j Ya w e
1
0
yy
y
B
G
y B
Stokes problem
2
0
0
v x p x x
v x x
v x x
B
B
• For media with obstacles
• Assumptions
20 1 2
20 1 2
, , ,
, , ,
v x v x y v x y v x y
p x p x y p x y p x y
Solution
• Comparing coefficients of the same order– Stokes equation:
– Conservation of mass:
– Boundary conditions:
10 0 0
00 1 0
0
: ,
: , , ,
y
y y x
p x y p p x
v x y p x y p x y
10 0: ,y v x y
0 1 0, , for v x y v x y y
for yB
for yB
• With we get for
• Separation of variables for both
where are solution of
0 0jx j xjp x e p x
0 1 0
0 0
, ,
,jy y j xj
v x y p x y e p x
v x y
yB
10 0
1 0
,
,
j
j
j xj
j xj
v x y w y p x
p x y y p x
0
0
y j y j j
y j
j
w y y e y
w y y
w y y
B
B
and j jw
0 1 and v p
Darcy’s law
• Averaging velocity over
where is given by
B
10 0: , u v x y dy u K p x B
,ij i j
K k
ij jik w y dyB
Conservation of mass
• Term of order in conservation of mass
• Integration over yields
0
1 0 0, ,y xv x y v x y
1
1 1
0
,
, ,
x y
Y
u x v x y dy
v x y d y v x y d y
B
B
Proposition
Proposition 5: The homogenization of the Stokes problem is given by
Proposition 6: The tensor K is symmetric and positive definite
1 0 , u K p x u
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
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