Biomedical Engineering Reference
In-Depth Information
Figure  6.22  Results of the numerical modeling. Velocities are indicated by the arrows and
show a Poiseuille parabolic profile, except at the entrance of the channel where the flow is totally
established. A few flow lines for concentration have been plotted proving that the distance of capture
depends on the initial position of the particle.
ematical software such as MATLAB if the geometry of the computational domain
is simple (rectangle or cylinder). A finite volume method can be set up by using a
semi-implicit Crank-Nicholson discretization scheme.
Figure 6.23 shows the indices for r and z ; the discretized equation at the node
( i , j ) is
n
+
1
n
+
1
n
n
é
ù
n
+
1
n
æ
2
ö
c
-
c
c
-
c
c
-
c
r
i j
,
i j
,
i j
,
i j
,
j
i j
, 1
+
i j
, 1
+
+
U
1
-
ê
-
ú
ç
÷
2
D
t
D
r
D
r
R
ê
ú
è
ø
ë
û
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
n
+
1
é
ù
c
-
2
c
+
c
c
-
2
c
+
c
c
-
c
D
1
i j
,
i j
,
i j
,
i
+
1,
j
i
-
1,
j
i j
, 1
+
i j
, 1
-
i j
, 1
+
(6.79)
=
ê
+
+
ú
2
2
2
r
D
r
(
D
z
)
(
D
r
)
ê
ú
j
ë
û
é
n
n
n
n
n
n
n
n
ù
c
-
2
c
+
c
c
-
2
c
+
c
c
-
c
D
1
i j
,
i j
,
i j
,
i
+
1,
j
i
-
1,
j
i j
, 1
+
i j
, 1
-
i j
, 1
+
+
ê
+
+
ú
2
2
2
r
D
r
(
D
z
)
(
D
r
)
ê
ú
j
ë
û
with the precaution that on the centerline ( r = 0), the terms in 1/ r should be removed
(because
c
r ). More on the numerical algorithm for the solution of the advection-
=
0
diffusion equation may be found in [10].
The results for a concentration “burst” of particles have been plotted in Figure
6.24.
Figure 6.23  Schematic view of the computational nodes and grid.
 
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