Biomedical Engineering Reference
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of curvature R , with an average velocity in axial direction u . The average transversal Dean velocity can
be estimated as [19]
y ¼ Re D h
R u
(6.6)
where D h is the hydraulic diameter, which is a characteristic transversal length scale. The ratio
between the axial time scale and the transversal time scale is
D h =y ¼ Re L
L
=
u
s axial
s trans ¼
(6.7)
R $
To complete a 180 turn, the axial and transversal time scales should match (
s axial /
s trans z
1), leading
to the required Reynolds number for a given radius of curvature R and length L :
R
L $
Re
z
(6.8)
Sudarsan and Ugaz [19] combined the characteristics of curved channel with geometric splitting to
realize sequential lamination in a simple planar design. There is no need of a complex three-dimensional
channel structure for geometric rotating and splitting, as shown in Section 4.3. The mixing concept is
depicted in Fig. 6.12 (b). Amixing unit consists of a single curved channel and multiple curved channels.
Fluid streams enter the mixer and flow through a 90 curve. The streams are simultaneously rotated by
90 at the end of the curved section, where they are split into multiple streams. These streams continue
the curved path for another 90 so that the fluid pairs are further rotated by 90 in their respective
streams. Lamination of multiple liquid layers is achieved, when the streams are finally rejoined. The
mixing unit is repeated for complete mixing. The simpler planar design reduces pressure drop and
mixing time compared to a conventional micromixer based on sequential lamination.
Vortices created by a sudden expansion were used in the design shown in Fig. 6.10 (b) to improve
transversal transport. These vortices can be combined with Dean vortices to create an even better
mixing effect. Sudarsan and Ugaz [19] reported the design depicted in Fig. 6.12 (c). This design
combines the Dean vortices in the vertical plane with the vortices of the sudden expansion in the
horizontal plane. The concept only works at a high Reynolds number (Re
32) where both Dean
vortices and expansion vortices become significant. The design parameters of this mixer design are the
position for the sudden expansion and the expansion ratio. Following (6.8) , a length of L
>
R /(2Re) is
required for a 90 rotation of the mixing fluids before the sudden expansion. Expansion ratios of 1:5
and 1:4 are common for reasonably large expansion vortices.
z
6.3 CHAOTIC ADVECTION AT LOW REYNOLDS NUMBERS
6.3.1 Chaotic advection with Dean vortices and complex 3-d channels
As mentioned previously, Dean vortices induce secondary flow that improves transversal transport
[19] . At lower Reynolds number (1
10), the small Dean numbers are large enough to create
chaotic advection in the mixing channel. The design reported by Sudarsan and Ugaz uses spiral
microchannels to induce Dean vortices in mixing channels for Reynolds numbers in the range
0.02
<
Re
<
<
Re
<
18.6 [20] . The spiral channel design allows the reduction of radius of curvature, thus, the
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