Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 20

where P 0 ¼ W 2 /( m 1 u 0 )d p /d x , representing the constant pressure gradient along the channel with the

reference velocity u 0 , and the ratio of viscosities b ¼ m 2 / m 1 . The no-slip conditions at the wall result in:

u 2 ð z Þ¼ 1 ; 0

u 1 ð z Þ¼

0 ; 0

u 2 y ;

h
2

u 1 ; ¼ 0

where h H / W is the dimensionless height of the channel. At the interface position r between the two

streams, the velocity and the shear stress are continuous:
u 2 ð

¼

z Þ¼ u 1 ð z Þ r ; r ;

y ¼r ¼ b vu 2 y ¼r :

vu 1
vy vy

1) and a constant fluid density, the interface position of the two streams
can be estimated, based on the mass conservation, as:

For a flat channel ( h <<

1

r ¼

bg :

þ

1

The solutions have the Fourier forms (0 < y * < 1, 0 < z * < h /2):

" z 2 #

N

h 2

= 4

u 1 ð y ; z Þ¼ P 0 cos qz ð A 1 cosh qy þ B 1 sinh qy Þ þ

2

n ¼ 1

" z 2 #

N

P 0
b

h 2

= 4

u 2 ð y ; z Þ¼ cos qz ð A 2 cosh qy þ B 2 sinh qy Þ þ

2

n ¼ 1

where q 1) p / h . The coefficients A 1 , A 2 , B 1 , B 2 can be obtained by solving the Fourier series

with the previous boundary conditions:

¼ (2 n

4 h 2

n þ 1

A 1 ¼ð Þ

1

3 p 3

ð 2 n 1 Þ

sinh 2 rq þð cosh rq cosh q þ b sinh 2 rq

A 1 ð

1 bÞð 1 cosh qÞ 1 bÞ

B 1 ¼

cosh 2 rq b sinh 2 rq ð 1 b Þ sinh rq cosh rq cosh q sinh q ð Þ

cosh 2 rq b sinh 2 rq cosh 2 rq

A 1 ð

1 cosh q Þ sinh rq sinh q ð Þ

cosh 2 rq b sinh 2 rq ð 1 b Þ sinh rq cosh rq cosh q sinh q ð Þ

cosh 2 rq b sinh 2 rq

A 1 ð

1 b Þ sinh rq cosh rq cosh q sinh rq ð Þ

A 2 ¼

cosh 2 rq

b sinh 2 rq

ð b Þ

sinh rq cosh rq cosh q sinh q

ð Þ

1

b sinh 2 rq cosh 2 rq

A 1

b cosh q þð b 1 Þ cosh rq cosh q þ

þ

cosh 2 rq b sinh 2 rqÞ ð 1 bÞ sinh rq cosh rq cosh q sinh qð