Micromixers based on molecular diffusion - Micromixers Fundamentals Design and Fabrication - page 175

y p in the polar coordinate system as shown in Fig. 5.11 , the
boundary condition of the solute flow is, after Fick's law:
d c
d r

x 2 Defining the variable r ¼ þ

r¼R ¼

D ¼ m 2

j

(5.24)

2 pRHD

where H is the height of the mixing chamber above the injection nozzle. Assuming a small mixing ratio
( m 2 m 1 ), the following boundary condition is acceptable for (5.23) :
d

r r¼N ¼

d

0 : (5.25)

Equation (5.23) can be rewritten for the polar coordinate system as
d 2

u
2 D

2

G
d r 2 þ

1
r

d d r

G ¼ 0 : (5.26)

The solution of (5.26) is the modified Bessel function of the second kind and zero order:
G ¼ K 0 ½ ur =ð

2 D

Þ:

(5.27)

The solution of (5.21) with the previously mentioned boundary conditions is

m 2 D K 0 ½ ur =ð 2 D Þ exp ½ ur cos q =ð 2 D Þ

Þ¼ _

pRH u 1

c ð r ; q (5.28)

K 1 ½ uR =ð

2 D

Þ K 0 ½ uR =ð

2 D

Þ

cos q

½ uR cos q =ð

2 D

Þ

exp

where q is the angular variable of the polar coordinate system.

FIGURE 5.12

Dimensionless concentration distribution around the injection nozzle: (a) Pe ¼ 0.1 and (b) Pe ¼ 1.