Biomedical Engineering Reference
In-Depth Information
Eqn
(2.66)
reduces to:
D
v
2
c
0
D
v
2
c
v
2
c
0
vx
vc
vt
þ
u
vc
u
0
vy
2
¼
vx
vx
2
þ
(2.69)
with
u
1
3
y
2
2
h
2
u
0
ð
y
Þ¼
2
:
(2.70)
Next,
(2.69)
can be integrated with respect to
y
:
y
vc
u
y
vc
vx
þ
D
v
2
c
vx
2
D
vc
0
y
3
2
h
2
u
vc
vy
¼
vt
þ
vx
þ
2
C
0
ð
x
;
t
Þ:
(2.71)
The symmetry and wall conditions imply that
C
0
(
x
,
t
)
¼
0 and
vx
D
v
2
c
vc
vt
þ u
vc
vx
2
¼
0
;
(2.72)
respectively. Integrating
(2.71)
with respect to
y
results in:
y
2
4
vc
vx
þ
y
4
8
h
2
u
D
c
0
¼
C
1
ð
x
;
t
Þ:
(2.73)
C
1
(
x
,
t
) can be determined by solving the condition:
1
h
Z
h
c
0
ð
x
;
y
;
t
Þ
d
y
¼
0
:
(2.74)
0
The final expression for the fluctuating component of the concentration is:
y
2
4
vc
vx
:
y
4
8
h
2
7
h
2
120
u
D
c
0
¼
(2.75)
Substituting
(2.75)
into
(2.64)
leads to:
vc
vt
þ
D
v
2
c
vx
2
¼
2
h
2
u
2
105
D
u
vc
vc
þ
0
:
(2.76)
Thus, the dispersion coefficient in a Poiseuille flow between two parallel plates is:
2
h
2
u
2
105
D
;
D
¼ D þ
(2.77)
where 2
h
is the gap between the two parallel plates.
2.3.2
Three-dimensional analysis
For axisymmetric channel geometry, such as a cylindrical capillary, the two-dimensional analysis
described in the above subsection is appropriate. For real channel geometry, the cross-sectional
velocity profile and, consequently, the dispersion coefficient also depend on the channel shape. In other
words, the second transversal spatial dimension needs to be considered in the analysis.
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