Biomedical Engineering Reference
In-Depth Information
b ¼
b
/
W
can be adjusted by the flow rate ratio between the sheath streams and the focused streams.
Considering the above effective diffusion coefficient in axial direction
(5.47)
and the molecular
diffusion coefficient in transversal direction, the analytical model for combined hydrodynamic
focusing and sequential segmentation in the domain depicted in
Fig. 5.16
(a) can be described by
a transient two-dimensional transport equation:
D
*
v
2
c
D
v
2
c
vy
2
vc
vt
þ
u
vc
vx
¼
vx
2
þ
(5.48)
where
c
is the concentration of the solute. The mixing channel is assumed to be two dimensional
with width
W
. The width of the focused stream is
b
(
Fig. 5.16
(a)). The transient inlet conditions at
(
x
0) are depicted in
Fig. 5.16
(b) and (c). This condition can be formulated as a function of both
y
and
t
:
¼
cðx ¼
0
Þ¼rðyÞsðtÞ
(5.49)
where
8
<
0
;
0
y
<
W
=
2
b
=
2
r
ð
y
Þ¼
1
;
W
=
2
b
=
2
y
W
=
2
þ
b
=
2
(5.50)
:
;
W
=
þ
b
=
y
<
W
0
2
2
and
8
<
c
0
;
0
t
aT
=
2
s
ð
t
Þ¼
1
;
aT
=
2
t
T
aT
=
2
(5.51)
:
c
0
;
T
aT
=
2
<
t
T
where
T
is the time period of the switching process and
a
is the switching ratio. The switching
frequency is then
f
¼
1/
T
. The final mixed concentration can be adjusted by both the focusing ratio
b
b
/
W
and the switching ratio
a
. The expected concentration at the end of a long mixing channel is
c
(
x ¼N
)
¼ abc
0
.
Normalizing spatial variables by the segment length
L ¼ uT
, the time by the switching period
T
,
and the concentration by the initial solute concentration
c
0
:
x
*
¼
¼ x
/(
UT
),
y
*
¼ y
/
UT
,
W
*
¼W
/
UT
,
t
*
¼ t
/
T
,
c
*
¼ c
/
c
0
, the transport Eqn (5.48) has the dimensionless form:
g
v
2
c
*
v
2
c
*
vy
*
2
vc
*
vt
*
¼
vc
*
vx
*
1
Pe
vx
*
2
þ
(5.52)
where
g
is the ratio of the effective diffusion coefficient
D
* in axial direction to the molecular diffusion
coefficient
D
in transversal direction:
D
*
D
¼
H
2
U
2
210
D
2
g
¼
1
þ
(5.53)
U
2
T
/
D
is the Peclet number defined based on the segment length
L
. Because the
concentration field at the entrance does not meet the condition of the Taylor-Aris dispersion model, we
and Pe
¼
UL
/
D
¼
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