Biomedical Engineering Reference
In-Depth Information
Figure 6.20
The length
L
is the same regardless of
R
if the flow rate is identical in both cases.
2
¶
c
¶
c
=
D
(6.71)
2
t
¶
¶
y
A normalized uniform
y
-distribution of the particles at the channel entrance can
then be expressed by the Fourier series expansion
¥
k
æ
ö
1
( 1)
-
æ
1
ö
π
y
å
0
(6.72)
=
cos
k
+
ç
÷
ç
÷
è
ø
2
æ
1
ö
è
2
w
ø
π
k
+
k
=
ç
÷
è
ø
2
Upon substitution of (6.72) in (6.71) and integration, one finds the solution
¥
k
( 1)
-
æ
1
π
y
ö
æ
ö
å
c y t
( , ) 2
=
c
cos
k
+
ç
÷
ç
÷
0
1
è
ø
æ
ö
è
2
w
ø
k
=
0
π
k
+
ç
÷
è
ø
2
(6.73)
æ
2
ö
2
2
1
π
æ
ö
exp
-
k
+
Dt
ç
ç
÷
÷
è
ø
2
w
è
ø
Integration of (6.73) with respect to
y
produces the time dependence of the concen-
tration
2
¥
æ
ö
k
2
( 1)
-
1
π
æ
ö
å
C t
( ) 2
=
c
exp
-
k
+
Dt
(6.74)
ç
÷
ç
÷
0
2
è
ø
2
2
w
æ
1
2
ö
è
ø
k
=
0
2
π
k
+
ç
÷
è
ø
Substitution of
t
by
x
/
U
brings back to the fix Eulerian coordinate system. We
note that each mode is attenuated with the axial distance
x
by the factor
2
æ
ö
k
2
C t
( )
( 1)
-
1
π
x
æ
ö
k
a
=
=
2
exp
-
k
+
D
ç
÷
ç
÷
k
2
è
ø
2
c
2
U
w
æ
1
2
ö
è
ø
0
2
π
k
+
ç
÷
è
ø
After some distance, all the modes are damped except the first mode correspond-
ing to
k
= 0. The persistence of the first mode only is sketched in Figure 6.21.
The attenuation in the
x
-direction is then
