Biomedical Engineering Reference
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where
s
el,0
is the average conductivity of both mixing streams. According to Lin et al.
[39]
, the critical
Rayleigh number is on the order of 1000, if:
3E
el
H
2
Dm
:
Electric Rayleigh number shows the ratio between transverse transport caused by electroviscous
velocity and molecular diffusion. A larger Rayleigh number means better mixing due to stronger
instability. Furthermore, the electric Rayleigh number is proportional to the square of the field strength
ð
Ra
el
¼
E
el
Þ:
Thus, mixing based on electrokinetic instability is sensitive to the applied voltage.
Ra
el
f
7.5.2
Instability caused by variation of electric field
Besides using a gradient in conductivity, instability can be induced by a periodic transversal flow
similar to the case of hydrodynamic instability depicted in
Fig. 7.3
. Similar to the hydrodynamic case
(
Fig. 7.11
), a transversal disturbance flow can be introduced using one or more disturbance channels.
The main mixing streams can be either pressure driven
[34]
or electrokinetic
[36]
.
For a simple two-dimensional electrokinetic disturbance flow, the Navier-Stokes equation in
y
direction reduces to:
m
v
2
y
r
vy
vt
¼
vx
2
:
(7.49)
If the Debye length is assumed to be very small compared to the channel width, a slip boundary
condition with Smoluchowski velocity:
3zE
el
;
0
m
y
wall
¼
exp
ð
iut
Þ
(7.50)
where
E
el,0
exp(
iut
) with the angular disturbance frequency
u
represent the applied disturbance
electric field. For the geometry depicted in
Fig. 7.11
and introducing the dimensionless variables
x
*
¼
x
/
W
x
,
t
*
¼
tu
and
y
*
¼
ym
/(
3zE
E0
) the solution for the disturbance velocity is
[34]
:
sinh
b
Þ
þ
sinh
b
x
*
x
*
ð
1
þ
i
Þð
1
ð
1
þ
i
Þ
y
*
x
*
t
*
it
*
ð
;
Þ¼<
exp
ð
Þ
(7.51)
sinh
½
b
ð
1
þ
i
Þ
where
R
and
i
stand for the real part and the imaginary unit, respectively. The ratio
b
is defined as:
W
x
2
m
b
¼
p
:
(7.52)
=
ru
r
p
is called the Stoke's penetration depth. Instability was observed by Oddy
et al.
[34]
for
b
on the order of 0.1 to 10. In a weak electric field strength, the flow described by
equation
(7.51)
is stable.
Figure 7.19
shows the typical dimensionless time-dependant velocity
profile at
b
The term
2
m
=
¼
5. At a high electric field, the flow becomes unstable with a three-dimensional chaotic
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