Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 40

The trajectories of a chaotic three-dimensional flow are complicated. Three-dimensional positions
of fluid particles can be reduced into a two-dimensional map called the Poincare´ section . In a time-
periodic system, Poincare´ section is a collection of intersections of trajectories with a plane. The
continuous trajectories become discrete points of the transformations P n /

P nþ 1 . The time needed
between the two points P n and P n þ 1 does not need to be the period of the system. In three-dimensional
space-periodic systems, the plane is taken at the same position of the repeated spatial structure. A
trajectory intersects all these periodic planes at several points. The collection of these points forms the
Poincar´ section. In this case, the transformation P n /

P n þ 1 is the advection cycle.

2.4.2 Examples of chaotic advection
2.4.2.1 Lorentz's convection flow
For a three-dimensional system, the equations in (2.85) are more than enough to create a nonintegrable
or chaotic dynamics. Lorenz [27] derived a simplified system of equations for convection rolls in the
atmosphere:

8
<

d x

=

d t

¼ ð y x Þ

Pr

d t ¼ x Ra

Ra c z

d y= y (2.89)

:

d z = d t ¼ xy bz

where the variable x is proportional to convective intensity, y is proportional to the temperature
difference between descending and ascending currents, and z is proportional to the difference in
vertical temperature profile from linearity in this system of equations. Pr, Ra, Ra c , and b are the Prandtl
number, Rayleigh number, critical Rayleigh number, and the geometric factor. Figure 2.15 shows the
solution of (2.89) for different Rayleigh numbers. A small change in Rayleigh number leads to a large
change in the solution.

2.4.2.2 Dean flow in curved pipes
The flow field inside a curved pipe was first derived by Dean [28] . For a more detailed review on
flow in curved pipes, see [29] . The following detailed derivation was given by Gratton [30] . The
model for the flow in a toroidal pipe is depicted in Fig. 2.16 . The pipe has the form of a toroid of
a radius of R . The pipe diameter is a . The coordinate system for this model is based on the cylin-
drical coordinate, where s is the coordinate of the toroid's center line q . The metric of this coordinate
system is:

1

R sin q 2

r

2 2 2 2 2

ð d q Þ ¼ þ ð d s Þ þð d s Þ þð r Þ ð d q Þ : (2.90)

With the assumption of a laminar flow, the change in s is zero. With u , v , and w are velocity
components in s , r , and q . Continuity Eqn (2.11) and Navier-Stokes Eqns (2.14) have the following
forms in the new coordinate system:
vy
vr þ

vw
vq þ

y
r þ

y sin q þ w cos q

1
r

¼ 0 ; (2.91)

R þ r sin q