Biomedical Engineering Reference
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where
D
d
is the diameter of the generated droplet. Initially, the interfacial tension is large enough to
keep the small droplet at the injection port. At the detachment moment, the continuous droplet growth
makes the drag force large enough to release the droplet. Combining the above equations results in the
droplet diameter
s
C
S
C
D
D
i
s
D
d
¼
2
r
c
u
c
$
(6.19)
Substitute
(6.17)
, with Re
¼ r
c
uD
c
/
m
c
into
(6.19)
results in:
s
s
m
c
D
c
D
i
u
c
$
D
d
¼
2
(6.20)
gz
u
c
=
D
c
:
Because the shear rate can be estimated as
The relation between droplet size and the
interfacial tension, and the shear rate of a T-configuration can be estimated as
r
s
m
c
g
$
D
d
f
(6.21)
Figure 6.30
shows the velocity field inside a droplet formed at a T-junction. The velocity field was
measured using microparticle image velocimetry (see Chapter 8). The formation process illustrated in
Fig. 6.30
is clearly in the shearing regime. The shearing regime is possible because of the small ratio
between the widths of the injection channel and the carrier channel. The high shear stress deforms the
droplet and stretches along the flow direction, allowing a large gap between the droplet and the channel
wall. Since the carrier fluid has enough place to flow through this large gap, the pressure drop across
the droplet is small and can be neglected in the force balance. The shear stress caused by the carrier
fluid leads to a single vortex in the droplet during formation (
Fig. 6.30
(a-g)) (micro-PIV). The single-
vortex flow pattern subsequently changes to a flow pattern with two vortices. These flow patterns inside
a droplet can be used for generating chaotic advection, as elaborated later in Section 6.4.2.
6.4.1.2 Passive bubble formations
In the case of bubble formation, the capillary number is usually extremely small due to the high gas/
liquid surface tension and the smaller shear rate due to the low viscosity of the gaseous phase. Thus,
shear force is negligible compared to surface tension force. A stable droplet formation process can
only be achieved with confined geometry, such as the T-configuration and the flow-focusing config-
uration, as shown in
Fig. 6.31 [41]
. According to Garstecki et al.
[42]
, the bubble diameter generated
by the flow-focusing configuration can be adjusted by the supply pressure
p
of the gas phase and the
flow rate
Q
of the liquid phase:
3
r
p
Qm
D
bubble
f
(6.22)
where
m
is the viscosity of the liquid phase. The formation process does not depend on the surface
tension.
In a T-configuration, the two formation regimes are determined by the gap
g
between the bubble
and the top channel wall. If this gap is small
g W
, the formation process is in the squeezing regime.
If the gap is large
g
z
W
, the formation process is in the shearing regime. The formation process in
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