Biomedical Engineering Reference
In-Depth Information
Table 2.2
Analytical solution for velocity field inside a straight channel
Channel
Type
Solution
1
r
2
r
u
ðrÞ¼
2
Circle
0
d
u ¼
1
8
r
2
m
x
1
y
2
a
2
z
2
u
ðy; zÞ¼
2
Ellipse
b
2
a
2
b
2
a
2
þ b
2
d
dx
u ¼
1
4
m
a
2
þ b
2
a
2
b
2
ln
a
2
r
2
þ
a
2
b
2
ln
ða=rÞ
ln
Concentric
annulus
u
ðrÞ¼
2
ðb=aÞ
ða=bÞ
"
#
d
a
2
þ b
2
a
2
b
2
ln
ða=bÞ
u ¼
1
8
m
x
,
ð
1
Þ
n
1
1
192
a
p
5
b
N
N
u
ðy; zÞ¼
4
p
3
cosh
½ð
2
n
1
Þ
p
z=
2
a
cosh
cosh
½ð
2
n
1
Þ
p
y=
2
a
ð
2
n
1
Þ
3
tanh
½ð
2
n
1
Þ
p
b=
2
a
ð
2
n
1
Þ
5
Rectangle
1
½ð
2
n
1
Þ
p
b=
2
aÞ
n ¼
1
n ¼
1
(
)
N
u ¼
a
2
3
d
1
192
a
p
5
b
tanh
½ð
2
n
1
Þpb=
2
a
ð
2
n
1
Þ
5
m
x
n ¼
1
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