Problem 4.60
[Difficulty: 3]
Given:
Data on flow at inlet and outlet of pipe
Find:
Ratio of outlet to inlet momentum flux
Solution:
∫
⋅
=
A
x
dA
V
u
r
ρ
mf
Basic equation: Momentum flux in x direction at a section
Assumptions:
1) Steady flow
2) Incompressible flow
Evaluating at 1 and 2
mf
x1
U
ρ
⋅
U
−
π
⋅
R
2
⋅
(
)
⋅
=
mf
x1
ρ π
⋅
U
2
⋅
R
2
⋅
=
Hence
mf
x2
0
R
r
ρ
u
2
⋅
2
⋅
π
⋅
r
⋅
⌠
⎮
⌡
d
=
2
ρ
⋅
π
⋅
u
max
2
⋅
0
R
r
r
1
r
R
⎛
⎜
⎝
⎞
⎠
2
−
⎡
⎢
⎣
⎤
⎥
⎦
2
⋅
⌠
⎮
⎮
⎮
⌡
d
⋅
=
2
ρ
⋅
π
⋅
u
max
2
⋅
0
R
y
r
2
r
3
R
2
⋅
−
r
5
R
4
+
⎛
⎜
⎜
⎝
⎞
⎠
⌠
⎮
⎮
⎮
⌡
d
⋅
=
mf
x2
2
ρ
⋅
π
⋅
u
max
2
⋅
R
2
2
R
2
2
−
R
2
6
+
⎛
⎜
⎝
⎞
⎠
⋅
=
ρ π
⋅
u
max
2
⋅
R
2
3
⋅
=
Then the ratio of momentum fluxes is
mf
x2
mf
x1
1
3
ρ
⋅
π
⋅
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 Fall '07
 Lear
 Fluid Mechanics, Flux

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