Biomedical Engineering Reference
In-Depth Information
If a pressure difference of
D
p
is applied across the channel length of
L
, the flow rate is:
(
1
)
tanh
ð
n
X
¼N
H
3
W
D
p
12
mL
192
H
2
n
1
ÞpH
Q
¼
:
5
p
5
W
2
W
ð
2
n
1
Þ
n¼
1
For a microchannel with low aspect ratio (
h
¼
H
/
W
0), the flow rate can be estimated as:
/
H
3
W
p
12
mL
ð
D
Qz
1
0
:
630
hÞ:
Example 2.4
(
Velocity distribution of streams with different viscosities in a rectangular micro-
channel
). Two immiscible fluids flow side by side in a rectangular microchannel of width
W
and height
H
. The viscosity ratio and flow rate ratio of the two fluids are
b
¼ Q
2
= Q
1
, respectively.
¼
m
2
/
m
1
and
g
Determine the velocity distribution in this microchannel
[4]
.
Solution.
The fully developed flow in the microchannel is governed by Navier-Stokes equations:
v
2
u
1
vy
2
þ
v
2
u
1
vz
2
1
m
1
d
p
d
x
¼
v
2
u
2
vy
2
þ
v
2
u
2
vz
2
d
p
d
x
where
m
1
and
m
2
are the viscosities of the two streams and d
p
/d
x
is the pressure gradient along the
x-axis. Normalizing the coordinate system by the channel
W
(
y
*
1
m
2
¼
¼
y
/
W
,
z
*
¼
z
/
W
) and the velocity by
a reference velocity
u
0
(
u
*
¼
u
/
u
0
), the dimensionless Navier-Stokes equations for the regions 1 and
2in
Fig. 2.5
are:
v
2
u
1
v
2
u
1
P
0
vy
2
þ
vz
2
¼
v
2
u
2
v
2
u
2
P
0
b
vy
2
þ
vz
2
¼
FIGURE 2.5
Dimensionless model of two immiscible streams for estimating the velocity distribution inside the mixing channel;
only half of the channel cross section (hatched areas) is considered in the analytical model.
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