Biomedical Engineering Reference
In-Depth Information
where
Z
a
Z
a
1
a
d
y
1
a
c
0
ð
c
ð
x
;
t
Þ¼
c
ð
x
;
y
;
t
Þ
x
;
y
;
t
Þ¼
0
0
0
Z
a
Z
a
u
1
a
d
y
1
a
u
0
¼
u
ð
y
Þ
:
0
0
0
Substituting
(2.62)
into
(2.60)
results in:
vc
vx
þ
D
v
2
c
v
2
c
0
vx
2
þ
v
2
c
vy
2
vc
0
vt
þð
vc
0
vx
vc
vt
þ
u
0
Þ
u
þ
¼
vx
2
þ
(2.63)
with
y¼
0
¼
vc
0
vy
Symmetry condition :
0
y¼h
¼
vc
0
vy
Wall condition :
0
c
t
j
t¼
0
¼
Initial condition :
c
þ
c
0
ð
x
;
y
Þ:
Averaging
(2.63)
from 0 to
h
leads to:
u
0
vx
d
y
Z
h
u
vc
0
vc
0
D
v
2
c
vc
vt
þ
1
h
vx
þ
¼
vx
2
:
(2.64)
0
The initial condition of the above equation is:
Z
h
1
h
c
j
t¼
0
¼
f
ð
x
;
y
Þ
d
y
:
(2.65)
0
Similar to Taylor's original approach, the dispersion coefficient can be derived from the above
equation if the fluctuation concentration
c
0
is known. Subtracting
(2.64)
from
(2.63)
leads to the partial
differential equation for
c
0
:
2
4
u
0
vx
d
y
3
u
0
D
v
2
c
0
Z
h
vc
0
vt
þ
u
vc
0
vc
0
vx
vc
0
vx
vc
0
v
2
c
0
vy
2
1
h
5
¼
u
0
vx
þ
vx
2
þ
(2.66)
0
with
Z
h
1
h
c
0
j
t¼
0
¼
c
0
ð
x
;
y
Þ
c
0
ð
x
;
y
Þ
d
y
:
(2.67)
0
Assuming that
vc
0
vt
;
vc
0
vx
;
v
2
c
vx
2
[
v
2
c
0
vx
2
;
vc
vt
[
vc
vx
[
(2.68)
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