Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 49

FIGURE 2.23
The basic unit of a twisted pipe consisting of two C-shaped sections.

vortices in the transverse cross-sectional plane can move fluid particles between the center of the
channel and its wall. A unit of the simplest configuration for chaotic advection is depicted in
Fig. 2.23 . The secondary flow in a twisted pipe causes the so-called Dean vortices. Fluid particles
rotate with an angle of c between successive units. The following analysis was reported by Jones

et al. [32] .
Using the polar coordinate system ( q , r ) in the transverse x-y plane, the stream function j and the
axial velocity u are determined through the following dimensionless equation system:

8
<

vj
vr

vu
vq

vj
vq

vw
vr

1
r

2 u ¼

V C

vj
vr

2De 2 u sin q
r

(2.114)

:

1
r

v
vq

vj
vq

v
vr

vu
vq

cos q vu
vr

2 j

2 j

V ¼ V þ ;

where De is the Dean number, a is the pipe radius, and R is the radius of curvature of the C-shape bend.
The Dean number describes the ratio between the centrifugal force and the viscous force. From
(2.109) , the Dean number is proportional to the Reynolds number. The dimensionless pressure gradient
is defined as:

a 2
Rum

vp
vq ;

C ¼ (2.115)

where u is the average axial velocity. In ( 2.114) , the lengths and velocities are normalized by the pipe
radius a and the average axial velocity u , respectively.