Micromixers based on molecular diffusion - Micromixers Fundamentals Design and Fabrication - page 180

The symmetry condition at the z *-axis is
vu 1
vy *

y * ¼ 0 ¼

0 : (5.42)

At the interface between the sample flow and the sheath flow, the velocity and the shear rate are
continuous:

8
<

u 2 r z * ¼ u 1 r z *

; ;

y * ¼r ¼ b vu 2

y * ¼r :

vu 1
vy *

(5.43)

:

vy *

For a flat channel ( h 1), the position of the interface can be estimated as

1

r ¼ (5.44)

þ

2 bk

1

where k m 1 is the flow rate ratio between the sheath streams and the mixing streams. A Fourier

analysis with the above boundary conditions results in the coefficients of (5.40) :

¼ _ m 2 =

b sinh qr cosh 2 qr cosh q sinh 3 qr cosh q sinh 2 qr sinh q cosh q A 1 ¼ A 0 ½ ð b 1 Þ

sinh 2 qr ð b 1 Þ ð cosh q cosh qr Þþð b 1 Þ sinh qr cosh qr

b cosh 2 qr sinh qr cosh 2 q sinh 3 qr cosh 2 q cosh q ð cosh q cosh qr Þ=½

sinh 2 qr cosh qr sinh q cosh q

ð b Þ

1

b cosh 2 qr cosh q sinh 2 qr cosh q

A 2 ¼ A 0 ½ ð b 1 Þ sinh qr cosh qr sinh q

(5.45)

ð cosh q cosh qrÞ=½b cosh 2 qr sinh 2 q sinh 2 qr cosh 2 q

ðb 1 Þ sinh qr

ðb 1 Þ sinh qr cosh qr sinh q cosh q

B 1 ¼ 0

Þ=½b cosh 2 qr cosh q sinh 2 qr cosh q B 2 ¼ A 0 ½ðb 1 Þ sinh qr ð cosh q cosh qr

ð b 1 Þ sinh qr cosh qr sinh q

4 h 2

n þ 1

where A 0 ¼ð

Þ

3 p 3 . Figure 5.15 shows the typical dimensionless velocity distribution
inside the flow channel for the same flow rate in all streams ( k

1

ð 2 n 1 Þ

1). The theoretical model is depicted
with the solid lines in Fig. 5.15 . The velocity of the sample flow is lower if the sample flow is more
viscous than the sheath flow ( b

¼

< 1) ( Fig. 5.15 (a)). If the sheath flows are more viscous ( b > 1), the

sample flow is faster ( Fig. 5.15 (b)).

5.5.3 Combination of hydrodynamic focusing and sequential segmentation
Both hydrodynamic focusing and sequential segmentation can be combined to improve convective/
diffusive mixing in a microchannel [29] . The concept reduces mixing paths in both transversal and
axial directions. Figure 5.16 describes this mixing concept. The micromixer has four inlets and one