Micromixers based on molecular diffusion - Micromixers Fundamentals Design and Fabrication - page 171

FIGURE 5.8

Concept of sequential lamination.

where the dimensionless space variable x * and time t * are defined as

x

x * ¼ (5.11)

w ð t Þ

and

Z t

D
wðt 0 Þ

2 d t 0 :

t * ¼ (5.12)

0

t * 1/2 . Thus, the relation for the The penetration distance d in the ( x *, t *) space is given by dx * w

penetration distance d x in the ( x , t ) space is

D
W 2 2 a ð

1
2

d x
W exp Þ ¼

exp 2 at 1 Þ : (5.13)

ð at

Mixing is complete if d x ¼ W final when the diffusive penetration distance is on the same order of the

striation thickness:

D
W 2 2 a ð

1
2

1 ¼ exp 2 at 1 Þ : (5.14)

The value of a is inversely proportional to the shear rate afu=W ; thus, Pe W f aW 2 / D . At the fully
mixed state [exp(2 at final )

[ 1], the required channel length of the mixer would be

L mixer
W f

ln Pe W : (5.15)

The transformations described above can be achieved either by sequential lamination or by chaotic
advection. Sequential lamination can be implemented by forced splitting and lamination or by chaotic