Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 43

C

v v 2

h A

v w
vs

u i

v 2
vq 2

1
r

1
r

v
vr

1
r 2

a 2 r 2

2 Ar

y ~ ¼ð r3 sin q Þ þ þ

vr 2 þ

ð Þþ~

1

(2.97)

þv3 v

Aða 2

cos q

v
vq Aða 2

1
r

1
r

r 2 r 2

vr þ

Þ sin q þ Þ

v w !

vp
vr vr þ w

vy
vq

1
r

v
r

v
vq

1
r

3A 2 a 2 r 2

2 sin q ¼

ð Þ

r

; (2.98)

v ~ !

v ~ p

vq þ vr þ ~

w w

r

vy
vq

1
r

1
r

v v
vr

1
r

3A 2 a 2 r 2

2 cos q

ð Þ ¼ ; (2.99)

where 3

a / R is the ratio between the pipe diameter and the radius of curvature, and all the higher
order of the tilde-marked terms as well as of 3 are neglected. Setting the small terms in (2.97) to zero
0

¼

¼ C 2 Av 2 Av (2.100)

and rearranging for C :

C ¼ 4 Av : (2.101)

Equation (2.97) then has the form:

v v 2

!

v 2

vp
vs

u
vr 2 þ

vu
vr þ

u
vq 2

1
r

1
r

1
r 2

2 Ar y ¼ 6 3vAr sin q þ : (2.102)

Separating variables according to r and q and substituting the separated variables:
~ u ¼ b u ð r Þ

sin q

; ~ y ¼ b y ð r Þ

sin q

; ~ w ¼ b w ð r Þ

cos q

; ~ p ¼ r b p ð r Þ

sin q

(2.103)

in (2.96) , (2.98) , (2.99) , and (2.102) , normalizing the velocities by w ¼ C =ð 4 v Þ and spatial variables

by a and applying the no-slip boundary condition at r ¼ 0 result in the dimensionless velocity

components:

r 2 1 r 6

8
<

2 1

1152 r sin q 19

3 3

4 r sin q þ 3 Re

21 r 2 9 r 4 u ¼ þ

144 sin q 4

r 2 1 r 2 2

3 Re

: (2.104)

y ¼ ;

:

144 cos q 1

r 2 4 7 r 4 ;

3 Re 23 r 2 w ¼ þ

where Re ¼ wa = v is the Reynolds number. Using the Cartesian coordinates ( x ¼ r sin q , y ¼ r cos q ),

the three velocity components have the form:

8
<

h

y 2
r h 0 ð

d x
d t ¼

3 Re
144

ð r Þþ r Þ ;

d y
d t ¼

3 Re
144

xy
r h 0 ð

: (2.105)

r Þ

:

d s
d t ¼

r 2

2 ð 1 Þ