Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 60

2.6.1.4 Electrokinetic flow in a cylindrical capillary
Figure 2.32 shows the model of electrokinetic flow in a cylindrical capillary. The Navier-Stokes
equation and the Poisson-Boltzmann equation are formulated for the cylindrical coordinate system:

r d u
d r

r d

m 1
r

d
d r

1
r

d
d r

d r

¼ 3E el ; (2.147)

r d

sinh z e

1
r

d
d r

d r 2 zem N

3

¼

k B T J

:

(2.148)

For simplicity, the following dimensionless variables are introduced:

u
u eo

ze J

k B T ;

zez
k B T ;

r
R :

J ¼ z ¼ u ¼ r ¼

; (2.149)

where R is the capillary radius. Equations (2.147) and (2.148) then have the linearized ( Section 2.6.1.1 )
dimensionless forms:

r

d u
d r

1
r

d
d r

1
z J ;

¼ (2.150)

r R

l D

J
d r

1
r

d
d r

d

J ; ¼

(2.151)

R
l D

2

d 2 J

d r 2 þ

J
r ¼

1
r

d

J :

The dimensionless boundary conditions are:
d u
d r

r ¼ 0 ¼

u j r ¼ 1 ¼ 0 ; 0 : (2.152)

Solving (2.150) results in:

2

N

J 0 ð l n r Þ
J 1 ðl n Þ

2 ðR=l D Þ C n

l n

u ð r Þ¼ ; (2.153)

z

n¼ 1

FIGURE 2.32
Model of electrokinetic flow in a cylindrical capillary.