Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 62

FIGURE 2.33

Model for electrokinetic flow in a rectangular microchannel.

thermal energy. Using the above dimensionless variables, the Navier-Stokes equation and the Pois-
son-Boltzmann equation have their dimensionless forms:

D h
l D

2

v 2 u
vz 2 þ

v 2 u
vy 2 ¼

1
z

J ¼ YJ ; (2.161)

D h
l D

2

v 2 v 2 J

vz 2 þ

J
vy 2 ¼

J : (2.162)

The dimensionless number

D h
l D

2

1
z

Y ¼ (2.163)

describes the interplay between the electrokinetic force and the friction force. Using the dimensionless
boundary conditions

vu
z j z ¼ 0 ¼

vu
y j y ¼ 0 ¼

0 0

u j z ¼W=D h ¼ 0 ; u j y ¼H=D h ¼ 0

:

and solving (2.161) lead to the dimensionless velocity distribution of the electrokinetic flow:

WH N
n¼ 1

N

4Y D h ða n z Þ ðb m y Þ

cos cos

u ¼

b 2 m

a n þ

m¼ 1

(2.164)

Z z ¼W=D h Z y ¼H=D h

a n z Þ b m y ÞJ d z d y ;

cos ð cos ð

0 0

where

a n ¼ ð

2 n ÞpD h

2 W

1

for n ¼ 1 ; 2 ; 3 .

b n ¼ ð

2 m

Þ pD h

1

for m ¼ 1 ; 2 ; 3 .:

2 H