Biomedical Engineering Reference
In-Depth Information
The transport effects in the previous system are based on the electrolytic Ohmic model and can be
described by the following governing equations:
The species transport equation
D s el
D t ¼
2 s el
D
V
(7.42)
where s el is the conductivity of the fluid and D is the effective diffusion coefficients of the ions
according to (2.40).
The Poisson-Boltzmann equation
V $ ð
s el VJÞ¼
0
(7.43)
where
J
is the electric potential in the fluid.
The continuity equation
V $
v
¼
0
:
(7.44)
The Navier-Stokes equation:
r Dv
2 v
2
Dt ¼V
p
þ
m
V
þ
3
ðV
JÞVJ
(7.45)
where p is the pressure, r and m are the density and dynamic viscosity of the liquid.
The above governing equations can be used for the subsequent instability analysis of this system. For
the details of this analysis, readers may refer to the original work of Chen et al. [24] .
In the parallel lamination system depicted in Fig. 7.18 (a), the transversal perturbation of electro-
kinetic instability is proportional to exp(-2 K x x *), where K x and x * are the dimensionless growth rate
and streamwise location, respectively. The larger the growth rate, the better is mixing. Both variables
are normalized by the channel half-width W /2. Similar to the case of pressure driven disturbance,
the instability is characterized by the magnitude of disturbance and the dimensionless temporal
frequency
U
:
u W
u eo
U ¼
(7.46)
where u eo ¼ 3zE el / m is the electroosmotic velocity and u is the temporal frequency of the perturbed
interface. The magnitude of disturbance is represented by the dimensionless electric Rayleigh number,
which is the ratio between the transversal electroviscous velocity u ev and the effective ion diffusion
coefficient D :
g
2 W
d
3E el H 2
mD
u ev W
D ¼
1
Ra el ¼
(7.47)
g
þ
1
where g
s el,H / s el,L is the conductivity ratio of the two streams, d is the diffusion length. The ion
concentration distribution is assumed to be s el ( y )
¼
s el ; H
s el ; L
¼
s el, L þ
erfc (2 y / d ). The ratio
2
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