Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 44

For the s component, all the 3 terms are canceled from the expression of u in (2.104) . The functions
of h ( r ) and h 0 ( r ) are:

4 4

r 2 1 r 2 2

1

h ð r Þ¼ ;

r 2 :

(2.106)

2 r 1

r 2 3

3

h 0 ðrÞ¼

Further normalization of the time and s by Re 3 /144 and Re 3 /288 results in the dimensionless
velocity components:

8
<

y 2
r

d x
d t ¼

h 0 ð

h ð r Þþ r Þ

d y
d t ¼

xy
r

h 0 ð

: (2.107)

r Þ

:

d s
d t ¼

r 2

1

Figure 2.17 shows the dimensionless three-dimensional velocity field (2.107) of a pipe cross section
graphically. The effect of the centrifugal force can be observed clearly. The stream function can be
determined in the same way as shown in Example 2.9:

Z h

y 2 d x

Z h 0
r

xy dy

h

y 2 d y d x

ZZ v
vx

h 0
r

h 0
r

j ¼ þ þ þ

4 4

r 2 1 r 2 2

y

¼ ¼ yh : (2.108)

FIGURE 2.17
The dimensionless velocity field of a pipe cross section.