Biomedical Engineering Reference
In-Depth Information
The corresponding Reynolds number, in this case Re
245, is very large for a typical microscale
application. Such vortex-based mixers require velocities as high as 10 m/s and pressure up to 15 bars.
Vortices caused by inertial effects can be generated at moderate Reynolds numbers with turns and
geometrical obstacles. For instance, a simple 90 bend in the mixing channel can generate vortices at
Reynolds numbers above 10 [14] . Mixing is achieved with a single bend at Reynolds numbers higher
than 30. Obstacles such as structures on channel wall [17] or throttling the channel entrance [18] can
also induce inertial instabilities into the standard T-mixer design. More details about this mixer type
are discussed later in Chapter 5.
For simplification in modeling and characterization, most micromixers in this topic are assumed to
work with Newtonian fluids. As discussed previously, instability in Newtonian flows at high Reynolds
number is caused by the competing viscous force and inertial force. Because most micromixers used
for biochemical analysis work with relatively low flow rates, the Reynolds number would not be high
enough for instability. Flow instabilities at low Reynolds number can be achieved using another force
to replace the inertial force. In active micromixers, these forces are induced by external sources and are
discussed later in Chapter 7. Diluting a small amount of highly deformable polymers would introduce
elasticity to a fluid. This class of fluid is called viscoelastic fluid and belongs to the non-Newtonian
fluid family. The elastic forces caused by stretching and recoiling of the polymer molecules can work
against the viscous forces to induce instabilities.
The shear stress in viscoelastic fluid does not jump to zero after the disappearance of a driving
force. Due to its elastic property, the stress decays with a characteristic time called the relaxation time .
In macroscopic mixing devices, the relaxation time is much smaller than the characteristic residence
time of the flow. Thus, elastic effect does not affect to a great extent the overall flow behavior. In
microscale, the relaxation time and the characteristic residence time of the flow are of the same order.
Thus, elastic forces become dominant.
As discussed in Section 2.5, the elastic effect of viscoelastic fluid flow can be characterized by the
Weissenberg number, which is the ratio between the relaxation time and the characteristic residence time
(2.125). Flows in microchannels have typically low Reynolds numbers and high Weissenberg numbers.
The ratio between these two numbers is called the elasticity number (2.127). Because of the lowReynolds
number and the high Weissenberg number, elasticity numbers in microchannel can reach up to 100.
Groisman and Steinberg [19] use a mixer design with repeated circular turns to induce viscoelastic
instability ( Fig. 5.7 ). Because of the turn, instability is caused by a complex interplay between inertial,
viscous, centrifugal, and viscoelastic forces. With a cross-section of 3 mm
¼
3 mm, the mixing channel
is large for common micromixers. The viscoelastic fluid is a solution of 80 p.p.m polyacrylamide
(PAA, molecular weight of 1.8
10 7 ), 65% saccharose, and 1% NaCl in water. Good mixing was
reported at an axial distance of about 41 cm from the entrance. However, with the relatively sharp turns
W / R
3/4.5, mixing can be achieved with Dean vortices as well (see Section 6.1.3).
Gan et al. [20] utilized viscoelastic instability at a sudden contraction to improve mixing. Good
mixing was achieved at low Reynolds numbers (Re
¼
<
1). The device was made of silicon. The
m
m
m
microchannels have a depth of 150
m. The sudden constriction has dimensions of 1000
m: 125
m:
m
1000
m. The micromixer has the flow-focusing configuration. The fluid of the middle stream consists
of 1 wt% polyethyleneoxide (PEO) in 55 wt% glycerol water. The fluid of the side streams consists of
0.1 wt% PEO in water. For a total flow rate of 12 ml/h, the corresponding Peclet, Reynolds,
Weissenberg, and elasticity numbers are Pe ¼ 49.4 10 6 ,Re
5070,
respectively. The very large elasticity number shows that inertial forces are negligible compared to
¼
0.06, We
¼
278, and El
¼
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