Biomedical Engineering Reference
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f
7 10 3 Re 2
þ 0
:
058Re þ 0
:
84
ð 0
<
Re
<
10 Þ
Re
;
0
:
5 Þ¼
80 Þ :
(6.5)
0
:
18Re þ 0
:
41
ð 10
<
Re
<
It is apparent from the above relations that the strength of vortices decays along the axial direction and
is proportional to the Reynolds number. Good mixing can be achieved if the turns are repeated before
the Dean vortices disappear. The higher the Reynolds number, the stronger are the vortices and the
better the mixing. If the turns are on a plane, the vortices are symmetric. Repeating the turns out-of-
plane leads to the case of a twisted pipe, as discussed in Section 2.4.2. Thus, chaotic advection in three-
dimensional 90 turns can further decrease the operation range of the mixer to 10
30.
Liu et al. [12] reported a three-dimensional serpentine mixing channel fabricated in silicon and
glass. The channel was constructed as a series of C-shaped segments positioned in perpendicular
planes ( Fig. 6.9 (a)). The mixing channel was shaped by anisotropic etching and had a typical trape-
zoidal cross-section with a largest width of 300
<
Re
<
m. The three-dimensional
structure is formed by etching on both sides of a polished silicon wafer. The channels are covered on
both sides of the wafer by glass plates using adhesive bonding. Mixing was evaluated based on the
reaction between phenolphthalein and sodium hydroxide streams, which are introduced by syringe
pumps. Chaotic advection was induced by the twisted intersecting streamlines. At a Reynolds number
of 70, full mixing was achieved after only two C-shaped segments. For the same mixing length, the
three-dimensional chaotic mixer produces 16 times more reaction product than a straight channel and
1.6 times more reaction product than the planar design [12] .
Yi and Bau constructed three-dimensional 90 turns by laminating multiple layers of prefired
ceramic tapes. Optical access was realized by thermal bonding of two glass slides on top and bottom of
the ceramic stack. Full mixing was achieved at a Reynolds number of 20.
Vijayendran et al. reported a three-dimensional serpentine mixing channel fabricated in PDMS [13]
( Fig. 6.9 (b)). The channel was designed as a series of L-shaped segments in perpendicular planes. The
mixer was tested at the Reynolds numbers of 1, 5, and 20. Better mixing was achieved at higher
Reynolds numbers.
Chen and Meiners reported another complex channel design realized in PDMS [14] . The periodic
structure is formed by two connected out-of-plane L-shapes and measures about 400 300
m
m and a depth of 150
m
m
m. The
FIGURE 6.9
Micromixers based on chaotic advection at intermediate Reynolds numbers: (a) C-shape [12] ; (b) L-shape [13] ;
and (c) connected out-of-plane L-shapes [14] .
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