Active micromixers - Micromixers Fundamentals Design and Fabrication - page 246

7.2.2 Hydrodynamic instability
Figure 7.3 shows the model of a typical mixer configuration with pressure disturbance. The solvent and
solute flow side by side at the inlet at a constant mean velocity u 0 , while the transversal disturbance
flow is time dependent y * y 0 cos( ut ). The net transversal flow is zero. Thus the mean velocity at the

outlet of the mixer is still u 0 . The mixing effect in this mixer is called “spatiotemporal resonance”,
which was coined by Okkels and Tabeling [5] . Assuming a laminar flow at low Reynolds numbers and
a two dimensional model, following dimensionless control parameters are defined:

¼

y 0
u 0

a ¼

u W

u 0 :
The governing equation for the stability analysis is:
dx
d t ¼ u þ f ðtÞ v

U ¼

(7.19)

where the function f ( t ) t ). Solving the Navier-Stokes equation (2.14) with the model and

boundary conditions depicted in Fig. 7.3 and tracking the fluid particles according to equation (7.19)

¼ a cos( U

FIGURE 7.3

Model of an active mixer with pressure disturbance at the inlet. The main flow has a velocity u 0 while the

transversal disturbance velocity is time-dependent y* ¼ y 0 cos(ut).