Biomedical Engineering Reference
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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
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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