Micromixers based on molecular diffusion - Micromixers Fundamentals Design and Fabrication - page 184

should consider two asymptotic cases: no dispersion ( g ¼ 1) and dispersion far away from the entrance

( g [ 1). The dimensionless inlet condition is

c * x * 0 ¼ r * y * s * t *

¼ ð Þ ð Þ (5.54)

where r * and s * are

8
< y * W * bW *

0 ; 0 < = 2 = 2

r * y * ¼ W * bW * y * W * bW *

; = = = þ = (5.55)

1 2 2 2 2

:

W * bW * y * W * 0 ; = 2 þ = 2 <

and

8
<

t *

; < a =

1 0 2

s * t * ¼ t *

0 ; a = 2 1 a = 2 (5.56)

:

< t * 1 ; 1 a= 2 1 :

The boundary conditions at the channel walls are
vc *
vy *

y * ¼ 0 ¼ y * ¼W * ¼

vc *
vy *

0 : (5.57)

Using the method of separation of variables and the above conditions, the solution of (5.52) is
c * x *

exp 1
2 g

Pe x * exp i 2 npt *

q

Pe 2

N

; t * ¼ A 00 þ ; y *

A 0 n < þ 8 g Pe npi

n¼ 1

s

( exp " 1
2 g

Pe ! x * # cos mpy *

W *

)

4 mp
W *

2

N

Pe 2

þ A m 0 þ

m¼ 1

s

exp " 1
2 g

!

x * #

mp
W *

2

8
<

9
=

Pe 2

Pe þ 8 g Pe npi þ

N N

þ A mn <

:

cos mpy *
W *

;

exp i 2 npt *

n ¼ 1 m ¼ 1

(5.58)

with

A 00 ¼ ab

2 b sin ð npa Þ

A 0 n ¼

np

sin h mp
2

i

i

sin h mp
2

2 a
mp

(5.59)

A m 0 ¼ ð þ b Þ ð b Þ

1 1

sin h mp

i i

sin h mp

4 sin ð npa Þ

A mn ¼

2 ð

1 þ bÞ

2 ð

1 bÞ :

mnp 2

where m and n are integer numbers, i is the imaginary unit, and indicates the real component of

a complex number. Nguyen and Huang [29] fabricated a polymeric micromixer based on this

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