Fundamentals of mass transport in the microscale - Micromixers Fundamentals Design and Fabrication - page 32

FIGURE 2.11

Model for determination of Taylor dispersion in the Poiseuille flow between two parallel plates.

effective transport models for the transverse averages of solutes flowing through channels with more
general cross-sectional geometries and flow properties. Common techniques are the use of asymptotic
analysis [14-17] , the theory of projection operators [18] , the center manifold theory [19] . All the above
works only dealt with nonreactive problems. Johns and DeGance considered the influences of a system
of linear reactions upon Taylor dispersion [20] . Yamanaka and Inui used their projection operator
theory to solve problems involving a single irreversible reaction [21] . The following example by
Bloechle [22] demonstrates an intuitive approach similar to that of Taylor [10] . The approach is called
the mean-fluctuation method commonly used in turbulent flow.
Example 2.8 ( Taylor dispersion in Poiseuille flow between two parallel plates [22] ). Determine the
dispersion coefficient of a Poiseuille flow between two parallel plates with a gap of 2 h as depicted in
Fig. 2.11 .
Solution. Because of the symmetric geometry, only half of the model is considered (

N< x <N ,

0 < y < h ). The governing Eqn (2.22) reduces to the two-dimensional form of the parallel plates model

( t > 0):

D v 2 c

v 2 c
vy 2

vc
vt þ

vc
vx ¼

u ð y Þ

vx 2 þ

(2.60)

where u ( y ) is the velocity distribution [2] :

1

y 2
h 2

3 u
2

u ð y Þ¼ : (2.61)

The boundary and initial conditions for (2.60) are:

y¼ 0 ¼

vc
vy

Symmetry condition : 0

y¼ 0

vc
vy

Wall condition : h ¼ 0

Þ:
The concentration and velocity can be formulated as the sum of an average component and
a fluctuating component:

Initial condition : c j t¼ 0 ¼ c 0 ð x ; y

c 0 ð

c ð x ; y ; t Þ¼ c ð x ; t Þþ x ; y ; t Þ

(2.62)

u 0 ð

u ð y Þ¼ u þ y Þ