Biomedical Engineering Reference
In-Depth Information
Figure 5.4
Anisotropic media. The preferred direction of diffusion is the x-direction.
The diffusion constant
D
must be replace by a coefficient matrix [
D
] where
é
D
D
D
ù
11
12
13
ê
ú
[
]
=
ê
D
D
D
D
(5.14)
21
22
23
ú
ê
ú
D
D
D
ë
û
31
32
33
The mass flux is then anisotropic and Fick's law can be written under the form [4]
Jx
J
ì
ü
ï
ï
=
[ ]
D c
Ñ
(5.15)
í
ý
y
ï
ï
J
î
þ
z
Finally the diffusion equation is
2
2
2
¶
c
¶
c
¶
c
¶
c
¶
c
=
D
+
D
+
D
+
(
D
+
D
)
11
22
33
23
32
2
2
2
¶
t
¶ ¶
y z
¶
x
¶
y
¶
z
¶
c
¶
c
+
(
D
+
D
)
+
(
D
+
D
)
31
13
12
21
¶ ¶
z x
¶
x y
¶
if the
D
s are taken constant. We may rotate the axis in order to transform the rect-
angular coordinates (
x,y,z
) into the rectangular coordinates (
x,h,x
) characterizing
the principal axes of diffusion and the preceding equation becomes
2
2
2
¶
c
¶
c
¶
c
¶
c
=
D
+
D
+
D
(5.16)
1
2
3
2
2
2
¶
t
¶
ξ
¶
η
¶
ζ
It is possible to make the further transformation
D
D
D
ξ ξ
=
,
η η
=
,
ζ ζ
=
1
1
1
D
D
D
1
1
1
where
D
is arbitrary, to obtain
