Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 42

FIGURE 2.16

Model of Dean flow in a toroidal pipe.

v v 2 y

v 2 y
vq 2

w 2
r

u 2 sin q

y vy

w
r

vu
vq

1
r

vu
vr þ

1
r

vy
vr þ

y
r 2 þ

1
r 2

2
r 2

vw
vq

vr þ r sin q ¼ vr 2 þ R þ

sin q vy

1 cos q

r

vy
vq

w cos q
r

þ

vr þ R þ r sin q

#

sin q

2 ð

y sin q þ w cos q Þ

(2.93)

ð R þ r sin q Þ

" v 2 w
vr 2 þ

u 2 cos q v 2 w

vq 2

y vw

w
r

vw
vq

yw
r

vu
vq þ v

vw
vr þ

w
r 2 þ

vy
vq

1
rr

1
r

1
r 2

2
r 2

vr þ r sin q ¼ R þ

1
r

sin q vw

vr þ

cos q vw

vq

y
r

þ

R þ r sin q R þ r sin q

#
;

cos q
ðR þ r sin qÞ

2 ð

y sin q þ w cos q Þ (2.94)

where p is the pressure and v is the kinematic viscosity. Assuming that the radius of curvature is much
larger than the pipe diameter ( R [ a ). The solution of the above four equations can be derived based

on the Poiseuille flow:

<

a 2 r 2 u ¼ A ð Þþ~ u

y ¼ ~ y

(2.95)

w ¼ ~ w

:

r ¼ Cs þ p

p

r ;

where A and C are the constants for velocity and pressure gradient, respectively. The tilde-marked
variables are the small perturbations. Rewriting the equations and ignoring the small products of the
tilde-marked variables lead to:

v

vr þ ~

y ~ y

r þ

v w

vq ¼

~

1
r

0 ; (2.96)