Homogenization and porous media Heike Gramberg, CASA Seminar Wednesday 23 February 2005 by Ulrich...
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Transcript of Homogenization and porous media Heike Gramberg, CASA Seminar Wednesday 23 February 2005 by Ulrich...
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