Biomedical Engineering Reference
In-Depth Information
The coupling between the two equations is realized by Fick's law
d
G
= - Ñ
D c
(7.89)
w
d t
Equations (7.89) can be substituted in (7.88) and we obtain the value of the wall
concentration as a function of the wall concentration (and its derivative)
D c
Ñ
+
k c
G
w
on w
0
G =
(
k c
+
k
)
on w
off
The same remarks as in the preceding section can be made concerning the fluctuat-
ing equilibrium near the wall.
Numerical Approach
Numerical methods must be set up to solve such problems. If one has access to finite
element software, the numerical approach is straightforward. If not, and if the ge-
ometry of the microchamber is simple, a numerical formulation based on a finite dif-
ference approach can be set up using the following discretization based on the grid
defined in Figure 7.45. The method is very similar to that of the preceding section.
First, using a Crank-Nicholson semi-implicit scheme [15], the advection-diffusion
equation (7.87) can be discretized under the form
é
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
ù
n
1
n
+
c
-
2
c
+
c
c
-
2
c
+
c
c
-
c
D
i j
,
i j
,
i j
,
i j
,
i
+
1,
j
i
-
1,
j
i j
, 1
+
i j
, 1
-
ê
ú
=
+
2
2
D
t
2
ê
(
D
x
)
(
D
y
)
ú
ë
û
é
n
n
n
n
n
n
ù
c
-
2
c
+
c
c
-
2
c
+
c
D
i j
,
i j
,
i
+
1,
j
i
-
1,
j
i j
, 1
+
i j
, 1
-
ê
ú
+
+
(7.90)
2
2
2
(
x
)
(
y
)
ê
D
D
ú
ë
û
(
)
n
+
1
n
+
-
1
c
-
c
i j
,
i
1,
j
-
v
i j
,
D
x
Figure 7.45
Schematic view of the discretization grid.
