Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 45

FIGURE 2.18

Dean vortices: (a) secondary velocity field and (b) stream line of the secondary velocity field.

Figure 2.18 (a) shows the secondary velocity field of the pipe cross section. The inner side of the
torus is on the left. Centrifugal force causes the fluid to move outward. The streamlines depicted in
Fig. 2.18 (b) show the two vortices on the lower and upper half of the pipe.
The trajectories of the fluid particles are calculated using the velocity solutions (2.107) and
numerical integration with the Runge-Kutta method. Projecting the particle position on a single
two-dimensional cross section results in the Poincar´ section. Figure 2.19 shows the trajectories and
Poincar´ sections of the Dean flow where the s -axis is straightened for clarity. The results clearly
show that independent of the orientation seeding lines, the Poincar ´ sections follow the streamlines
as depicted in Fig. 2.18 (b). If this flow is used in a micromixer, the solvent and solute should be
introduced on the left and right of the cross section or the outer and inner side of the curved
channel ( Fig. 2.18 ), so that the trajectories of the fluid particle can sample both sides of the
channel. If the solvent and solute are introduced in the upper and lower halves of the channel, the
trajectories will keep them in their respective channel section and advective mixing will not work.
Even if the solvent and solute enter at the outer and inner side of the curved channel, the
trajectories are stable, elliptic, and homoclinic. Transversal transport is advective but not chaotic.
This means, they do not cross each other. Therefore, chaotic advection cannot be realized with the
original Dean flow.
The above analysis assumes a small ratio between the pipe diameter and radius of curvature, a /

R 1. For realistic channel designs, this ratio can be approximately unity, and the secondary flows are

more obvious. In this case, the flow is characterized by the Dean number:

r a

R

De ¼ Re (2.109)

where Re is the Reynolds number. The Dean number represents the ratio between centrifugal force and
the inertial force. There exists a critical Dean number De cr ¼ 150 where the secondary flow pattern

changes. For De < 150, there are only a pair of counter-rotating vortices as analyzed above. At higher

Dean numbers De > 150, the centrifugal force is dominant, leading to the formation of two additional