Biomedical Engineering Reference
In-Depth Information
Figure 6.18
(a) Experimental view of diffusing particles in a Poiseuille flow and in an
electro-osmotic flow. (b) Experimental velocity profile in both cases. Dispersion is reduced if the
diffusion front is flat.
2
2
æ
2
4
ö
¶
c
¶
c UR
¶
c
1
r
1
r
¶
c
=
+
- +
-
»
(6.60)
ç
÷
2
2
4
¶
x
¶
x
4
D
3
2
¶
x
¶
x
R
R
è
ø
To be satisfied, relation (6.60) requires
2
2
¶
c
UR
¶
c
>>
2
¶
x
4
D
x
¶
If
L
is the length over which a noticeable change in
c
can occur, we may ap-
proximate the gradients by
¶
c c
x L
»
¶
2
¶
c
c
»
2
2
¶
x
L
then the preceding inequality may be written as
LD
UR
(6.61)
>>
1
2
and, taking into account (6.59),
L
UR
>>
>> 7
(6.62)
R
D
To these two conditions, we add the condition for a laminar flow
U
F
Re
=
<<
2000
(6.63)
ν
The three conditions (6.59), (6.61), and (6.63) give the limits of applicability of
(6.57) for a cylindrical tube.
2
2
R
D
D
refers to cylindrical tubes only. Another value of
D
eff
can be obtained by the same reasoning for a flow limited by two parallel plates [2]
The value
=
eff
48
2
2
H U
D
=
(6.64)
eff
210
D
