Biomedical Engineering Reference
In-Depth Information
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
Search WWH ::




Custom Search