Problem 4.59
[Difficulty: 3]
CS
x
c
y
2
h
d
Given:
Data on flow at inlet and outlet of channel
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
−
2
⋅
h
⋅
(
)
⋅
w
⋅
=
mf
x1
2
ρ
⋅
w
⋅
U
2
⋅
h
⋅
=
Hence
mf
x2
h
−
h
y
ρ
u
2
⋅
w
⋅
⌠
⎮
⌡
d
=
ρ
w
⋅
u
max
2
⋅
h
−
h
y
1
y
h
⎛
⎜
⎝
⎞
⎠
2
−
⎡
⎢
⎣
⎤
⎥
⎦
2
⌠
⎮
⎮
⎮
⌡
d
⋅
=
ρ
w
⋅
u
max
2
⋅
h
−
h
y
1
2
y
h
⎛
⎜
⎝
⎞
⎠
2
⋅
−
y
h
⎛
⎜
⎝
⎞
⎠
4
+
⎡
⎢
⎣
⎤
⎥
⎦
⌠
⎮
⎮
⎮
⌡
d
⋅
=
mf
x2
ρ
w
⋅
u
max
2
⋅
2 h
⋅
4
3
h
⋅
−
2
5
h
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
=
ρ
w
⋅
u
max
2
⋅
16
15
⋅
h
⋅
=
Then the ratio of momentum fluxes is
mf
x2
mf
x1
16
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 Fall '07
 Lear
 Fluid Mechanics, Flux

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