Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 36

d with a continually varying height, the
dispersion coefficient is not determined by the channel height and aspect ratio. The channel width W is
the only geometric parameter that determines the dispersion coefficient. For instance, the dispersion
coefficients for channels with triangle, parabolic, and elliptical cross sections are:

Ajdari et al. [24] argued that for a shallow channel W [

0052 W 2 u

D triangle ¼ D þ 0 :

D ;

(2.80)

0031 W 2 u D parabolic ¼ D þ 0 :

D ;

(2.81)

0022 W 2 u D elliptical ¼

D þ 0 :

D ;

(2.82)

respectively.
In the case of an elliptical cross section, a low aspect ratio W [ d refers to an eccentricity of unity

3 ¼ 1. Substituting 3 ¼ 1in (2.79) results in a factor:

W 2
d 2 :

210
192

5
12

f ¼ (2.83)

Substituting (2.83) into (2.78) leads to the equation:

W 2 u
D :

5

D elliptical ¼ D þ (2.84)

192 12

Because 5/(192 12) z 0.0022, the approaches of Dutta [23] and Ajdari [24] agree in the case of
a shallow elliptical cross section.

2.4 CHAOTIC ADVECTION
2.4.1 Basic terminologies
The term chaotic advection refers to the phenomenon where a simple Eulerian velocity field leads to
a chaotic response in the distribution of a Lagrangian marker, such as a tracing particle [25] . Advection
refers to species transport by the flow. A flow field can be chaotic even in the laminar flow regime.
Chaotic advection can be created in a simple two-dimensional flow with time-dependent disturbance
or in a three-dimensional flow even without time-dependent disturbance. It is to be noted that chaotic
advection is not turbulent. For a flow system without disturbance, the velocity components of chaotic
advection at any point in space remain constant over time, while the velocity components of turbulence
are random. The streamlines of the steady chaotic advection flow across each other, causing the
particles to change their paths. Under chaotic advection, the particles diverge exponentially and
enhance the mixing between the solvent and solute flows. In a time-periodic system, the condition for
chaos is that streamlines cross at two consecutive time instants.
There are few terminologies related to visualization of an Eulerian velocity field. The first and most
common terminology is the pathline , also called trajectory of a fluid particle in the flow field. In
experiments, pathlines, orbits, or trajectories can be obtained by an image with a long-time exposure of
a fluorescent fluid particle.