Biomedical Engineering Reference
In-Depth Information
where
J
0
and
J
1
are the Bessel functions (of the first kind) of order 0 and 1, respectively.
l
n
is the
n
th
positive zero value of the Bessel function
J
0
(
l
n
)
¼
0.
C
n
is a function of the dimensionless potential
J
*:
Z
1
xJ
0
ðl
n
xÞJ
ðxÞ
C
n
¼
d
x:
(2.154)
x ¼
0
The boundary conditions for
(2.151)
are:
r
Þ
d
r
d
Jð
J
j
r
¼
1
¼
z
;
0
:
(2.155)
j
r
¼
0
¼
The solution for the dimensionless potential is:
I
0
R
l
D
r
I
0
R
l
D
J
ðr
Þ¼z
;
(2.156)
i
n
J
0
(
ix
) are the modified Bessel functions of the first kind and zero order. Substituting
(2.156)
into
(2.154)
results in:
where
I
0
(
x
)
¼
z
I
0
R
l
D
J
1
ð
l
n
Þ=
I
1
ð
R
=
l
D
Þ
C
n
¼
2
:
(2.157)
l
n
1
Rl
D
l
n
þ
The dimensionless velocity distribution in a cylindrical capillary is then:
R
l
D
2
4
3
5
:
2
R
l
D
2
I
1
I
0
N
l
n
r
Þ
J
1
ðl
n
Þ
1
l
n
J
0
ð
1
u
ðr
Þ¼
R
l
D
(2.158)
R
l
D
2
n
¼
1
l
n
þ
2.6.1.5 Electrokinetic flow in a rectangular microchannel
Due to the characteristics of microtechnology, many micromixers have a rectangular cross section.
Figure 2.33
shows the model of electrokinetic flow in a rectangular microchannel. The Navier-Stokes
equation and the Poisson-Boltzmann equation are formulated in the Cartesian coordinate system as:
m
v
2
u
3E
el
v
2
sinh
ze
v
2
u
vy
2
v
2
J
vz
2
þ
J
vy
2
2
zen
N
3
J
k
B
T
v
2
z
þ
¼
¼
:
(2.159)
Following dimensionless variables are introduced:
u
u
eo
;
ze
J
k
B
T
;
zez
k
B
T
;
z
D
h
;
y
D
h
;
u
¼
J
¼
z
¼
z
¼
y
¼
(2.160)
where
D
h
¼
4
WH
/(
W
þ
H
) is the hydraulic diameter of the microchannel. For simplicity,
the
approximation of sinh (
x
)
x
is used. As mentioned above, this assumption is correct if the hydraulic
diameter
D
h
is much larger than the Debye length
l
D
or the ion concentration is dilute. At the
molecular scale, this assumption means that the electric energy of the ions is much smaller than their
¼
Search WWH ::
Custom Search