Micromixers based on molecular diffusion - Micromixers Fundamentals Design and Fabrication - page 176

Introducing the Peclet number Pe ¼ 2 uR = D , the dimensionless radial variable r * ¼ r / R , and the

dimensionless concentration

c

c * ¼ (5.29)

2 m 2 =ð _ pH Þ

the dimensionless form of (5.28) is

K 0 Pe r *

4 Pe

c * r * q ¼

=

cos q exp

1 4 cos q

r * ; ½ Pe ð (5.30)

K 1 ð Pe = 4 Þ K 0 ð Pe = 4 Þ

where K 1 is the modified Bessel function of the second kind and first order. Figure 5.12 shows the
typical dimensionless concentration distribution around a single injection nozzle at different Peclet
numbers.

5.5 FOCUSING OF MIXING STREAMS
5.5.1 Streams with the same viscosity
Hydrodynamic focusing reduces the mixing path by decreasing the width of the solute flow. Two
solvent streams work as sheath flows in this concept ( Fig. 5.13 (a)). Knight et al. [10] reported detailed
experimental results of hydrodynamic focusing. However, the theoretical model reported in [10] was
erroneous. The correct model is presented as follows:
If the sheath streams have the same viscosity as the sample flow and all liquids are incompressible,
the effect of hydrodynamic focusing can be represented by a simple network model. Assuming that the
flow is laminar, the pressure difference across a microchannel is proportional to the flow rate.
Figure 5.13 (b) shows the network model of hydrodynamic focusing where microchannels are
represented by fluidic resistances. For simplicity, we further assume that both sheath microchannels
have the same fluidic resistance of R , which is defined as the quotient between the applied pressure
difference and the flow rate: R ¼ Dp= Q . The resistances of the inlet microchannel and the mixing

FIGURE 5.13

Focusing concepts: (a) geometric focusing and (b) hydrodynamic focusing.