Biomedical Engineering Reference
In-Depth Information
x
y
c
(6.36)
x
*
=
, *
y
=
, *
c
=
L
d
c
0
where L is a reference axial distance and c 0 is a reference concentration. Introducing
the Peclet number and taking into account (6.36), (6.35) can then be rewritten as
2
3
d
c
*
c
*
é
ù
2
(6.37)
P
-
y
=
1
*
e
ë
û
L
x
2
2
*
y
*
Now, we follow Levêques's approach, and change the vertical origin
y
1
= -
y
*
so that y is zero at the wall. Equation (6.37) becomes
2
3
d
c
*
c
*
2
P
[2
y
� �
-
y
]
=
e
2
2
L
x
*
y
2
y y y . If
we note δ , the reduced boundary layer thickness, and note that at a distance from the
wall δ , the advection and diffusion terms are of the same order, we obtain the scaling
In the boundary layer, the distance y is small and we can assume
ë 2
� �
-
»
2
û
c
c
-
c
*
*
w
0
»
x
c x
*
0
2
c
c
-
c
*
w
0
»
2
2
y
c
δ
0
and finally
d
3
3
P
δ »
1
e
x
Thus,
1
3
δ æ
x
ö
1
» ç
(6.38)
÷
1
3
d
è
3
d
ø
P
e
Take the case of Figure 6.11: d = 1 mm and Pe = 10,000. Equation (6.38) re-
duces to
1
3
δ »
0.32 x
d
and for x = 1 cm, δ » 0.0 d . This result agrees with the numerical result of Figure
6.11 and confirms the sketch of Figure 6.10.
To conclude, concentration boundary layers are often present in microfluidics
and they are to be taken into account for the comprehension and calculation of the
transport phenomena.
 
Search WWH ::




Custom Search