Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 47

FIGURE 2.20

Model of a helical pipe.

The flow in this coordinate system has only a second-order dependence on . Thus, the solution (2.107)

can be used for helical pipes by changing the basis to the new coordinate system (2.110) . The solution
for the velocity field is then:

s

8
<

h 0
r yx sin

h 0
r y 2 cos

d x
d t ¼

h cos ð ls Þþ ð ls Þþ ð ls Þ;

h sin

h 0
r xy cos

h 0
r x 2 sin

d y
d t ¼

ð ls Þþ ð ls Þþ ð ls Þ ; (2.111)

:

d s
d t ¼

r 2 1 ;

where the curvature and torsion are combined in the geometry parameter:

114 v
u

k ;

l ¼ (2.112)

where u is the mean velocity in s direction. The stream function of (2.111) has the form:

r 2
2

1

lr 2
2 4 4

r 2 1 r 2 2

y

j ¼ þ : (2.113)

Figure 2.21 shows the streamlines calculated using (2.113) . The initial positions of the particles in
the depicted trajectories are on a seeding line parallel to the y -axis. The results show that at increasing
torsion, the secondary flow transforms from two counter-rotating vortices into a single vortex.
The Poincar´ sections of flow helical pipes with different geometry parameters are shown in
Fig. 2.22 . The flow is initially sampled with a seeding line parallel to the y -axis. At l

0, there is no
torsion and the pipe is a torus. The flow is clearly not chaotic; the particles follow the streamlines. At
l

¼

> 0, chaotic advection is apparent.

2.4.2.4 Flow in twisted pipes [32]
While a straight channel is one dimensional, a three-dimensional flow (transverse cross-sectional
plane and longitudinal axis) can be created in curved channels. In such curved channels, secondary