Biomedical Engineering Reference
In-Depth Information
Q
dis
u
»
(4.85)
growth
ab
Remember that
b
is the height of the channel. On the other hand, the neck
shrinks at a rate
Q
cont
u
»
(4.86)
squeeze
ab
If
d
denotes the width of the neck, the time needed to achieve the squeezing is
approximately
d
d ab
τ
»
»
squeeze
u
Q
squeeze
cont
The total length of the droplet when it detaches is then
Q
dis
L a u
» +
τ
» +
a d
Q
growth squeeze
cont
If we note
a
=
d/a
, and scale by
a
, we nearly recover (4.84)
L
d Q
Q
dis
dis
» +
1
= +
1
α
(4.87)
a
a Q
Q
cont
cont
However, at this point,
a
is not a constant (
a
=
d/a
) whereas in (4.84)
a
is a
constant, with a value close to either 0 or to 1. In the case where
Q
dis
<
Q
cont
, we
have seen that
L ~ a
, which is equivalent to
d
= 0. In the case where
Q
dis
>
Q
cont
,
u
growth
>
u
squeeze
, which means that the growth velocity is larger than the squeeze
velocity. It is observed that the width of the neck
d
does not vary quickly during the
elongation phase; it suddenly goes to zero at the breakup. This is due to a little gap
between the blob and the wall (
e
in Figure 4.57) that vanishes suddenly at breakup.
Hence, the squeezing velocity is somewhat smaller than its value from (4.86). These
observations explain why the ratio
d/a
can be approximated by a constant
a
, of the
order of 1. Finally, we can approximate
d
»
α
H Q
-
Q
(
)
dis
cont
a
and (4.87) collapses to (4.84).
Droplets Formation: Frequency Control of the Droplet Size
According to (4.84), the size of the droplet is of the order of the channel width, in
any case always larger than the channel width. The question is: how can monodis-
persed droplets be produced smaller then the channel width? It has been found [61,
62] that smaller droplet sizes can be produced if the incoming rate of liquid was
modulated in frequency. Without frequency modulation, there is a natural frequency
f
0
of droplet detachment. By superposing a tunable forcing frequency, resonances
can be found leading to the formation of monodispersed droplets whose sizes differ
notably from that of (4.84). Such regimes are called synchronized regimes. In such
