Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 28

2.3 TAYLOR DISPERSION
Taylor dispersion is an effective mechanism for mixing a solute in a distributed velocity field, such as
a pressure-driven flow in a microchannel. This axial effect arises from a coupling between molecular
diffusion in the transverse direction and transverse distribution of the flow velocity. Figure 2.8 illustrates
the difference between molecular diffusion in a plug-like flow and Taylor dispersion in a distributed
flow. In a uniform flow field, such as the plug-like electroosmotic flow, advection and diffusion are
independent. Axial diffusion is the same as molecular diffusion ( Fig. 2.8 (a)). In a distributed flow field,
such as the pressure-driven flow with a parabolic velocity distribution, the solvent is stretched more in
the middle of the channel than near the wall, due to axial convective transport. The resulting concen-
tration gradient between the different fluid layers is then blurred by diffusion in the transverse direction
(see Fig. 2.8 (b)). As a result, the solute appears to be “diffusing” in the axial direction at a rate that is
much faster than what would be predicted by ordinary molecular diffusion.

2.3.1 Two-dimensional analysis
The dispersion coefficients are derived by a two-dimensional model, involving one axial and one
transversal spatial dimension. G. I. Taylor was the first to present a working model for the transverse
average that managed to capture the influences of both transverse diffusion and the transverse vari-
ations of the fluid velocity field. The analysis of Taylor [10] is based on the model of a long cylindrical
capillary with a radius r 0 ( Fig. 2.9 ). The derivation of the dispersion coefficient given below follows
Brenner and Edwards [11] . According to Table 2.2 , the velocity distribution in the capillary cross
section is:

2 1

r
r 0

2

2 u 1

r
r 0

2

u ð u ¼ r

u ¼ u ð r Þ (2.41)

/

FIGURE 2.8
Particle distribution in a microchannel with: (a) a uniform velocity profile and (b) a parabolic velocity profile.