Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 59

Equation (2.136) has then the homogenous form:

d 2 u

ð y Þ

¼ 0 : (2.140)

d y 2

With the boundary conditions:

y¼ 0 ¼

d u
d y

0

u j y¼h ¼ u eo ;

the solution of (2.140) is:

u eo Jð y z 1 u ð y Þ¼ : (2.141)

Introducing the dimensionless velocity u *, dimensionless potential J *, dimensionless zeta potential
z

*, and the dimensionless spatial variable y *:

u
u eo ;

ze J

k B T ;

zez
k B T ;

y
h :

u ¼ z ¼ y ¼

J ¼ (2.142)

The solution (2.141) and the Poisson-Boltzmann Eqn (2.132) have the dimensionless forms:

u ðy Þ¼ J ð

y Þ
z

1 ; (2.143)

h
l D

2

J 2
d 2 y ¼

d

J : (2.144)

The boundary conditions for (2.144) are:

J j y ¼ 1 ¼ z

J
d y j y ¼ 1 ¼

d

0 :

The solution of (2.144) is:

cosh h

l D y

J ¼ z

l D Þ :

(2.145)

ð h =

cosh

Substituting (2.145) into (2.143) results in the dimensionless velocity distribution of a electrokinetic
flow between two parallel plates:

cosh h

l D y

u ð y Þ¼

l D Þ

1 : (2.146)

cosh ð h =