Biomedical Engineering Reference
In-Depth Information
The first-order perturbation solution results in the following equations of the particle motion
[32]
:
8
<
h
y
2
h
0
ð
d
x
d
t
¼
a
1152
r
Þ
ð
r
Þþ
r
d
y
d
t
¼
a
1152
xy
r
h
0
ð
(2.116)
r
Þ
:
d
q
d
t
¼
1
4
bð
1
r
2
Þ:
De
C
2
, and
b
where
a
De
C
/Re.
Using the angle
q
to describe the three-dimensional motion of fluid particles, the velocity
components in
x-y
plane can be formulated as
[32]
:
d
x
d
q
¼
¼
¼
8
<
1152
b
4
7
y
4
a
5
x
2
23
y
2
x
4
8
x
2
y
2
þ
þ
þ
(2.117)
192
b
xy
3
y
2
:
:
d
y
d
q
¼
a
x
2
This solution results in chaotic advection for a combination of
a
/
b
and
X
. For instance, the most
chaotic pattern is achieved with
a
/
b
90
. The condition for chaotic advection to occur
¼
100 and
c
¼
in this configuration is:
2 arctan
sinh
pa
192
b
c
:
(2.118)
Figures 2.24-2.26
show the Poincar´ sections with the different model parameters.
2.4.2.5 Flow in a droplet
With the increasing popularity of droplet-based microfluidics, mixing in droplets becomes
a crucial task in designing a droplet-based lab-on-a-chip. The analytical solution for the internal
flow inside a droplet was first reported by Hadamard
[33]
. Consider a sp
he
rical microdroplet with
aradius
a
. The droplet experiences a uniform shear flow of a velocity
u
in the
z
-axis. We now
consider the viscosity ratio
b
m
1
/
m
2
,where
m
1
is the viscosity of the droplet fluid and
m
2
is the
viscosity of
t
he surrounding flui
d.
Normalizing the spatial variables by the droplet radius
a
,the
velocity by
u
and the time by
a
¼
=
u
result in the dimensionless equations of particle motion inside
the droplet
[34]
:
8
<
d
x
d
t
¼
zx
2
ð
1
þ
b
Þ
d
y
d
t
¼
zy
(2.119)
ð
þ
b
Þ
:
2
1
z
2
2
y
2
2
x
2
d
z
d
t
¼
1
:
2
ð
1
þ
b
Þ
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