Problem 4.106
[Difficulty: 4]
CS
x
y
a
b
d
c
F
f
Given:
Data on flow of boundary layer
Find:
Force on plate per unit width
Solution:
Basic equations: Continuity, and momentum flux in x direction
Assumptions:
1) Steady flow
2) Incompressible 3) No net pressure force
From continuity
ρ
−
U
0
⋅
w
⋅
δ
⋅
m
bc
+
0
δ
y
ρ
u
⋅
w
⋅
⌠
⎮
⌡
d
+
0
=
where m
bc
is the mass flow rate across bc (Note: sotware
cannot render a dot!)
Hence
m
bc
0
δ
y
ρ
U
0
u
−
()
⋅
w
⋅
⌠
⎮
⌡
d
=
For x momentum
F
f
−
U
0
ρ
−
U
0
⋅
w
⋅
δ
⋅
⋅
U
0
m
bc
⋅
+
0
δ
y
u
ρ
⋅
u
⋅
w
⋅
⌠
⎮
⌡
d
+
=
0
δ
y
U
0
2
−
u
2
+
U
0
U
0
u
−
⋅
+
⎡
⎣
⎤
⎦
w
⋅
⌠
⎮
⌡
d
=
Then the drag force is
F
f
0
δ
y
ρ
u
⋅
U
0
u
−
⋅
w
⋅
⌠
⎮
⌡
d
=
0
δ
y
ρ
U
0
2
⋅
u
U
0
⋅
1
u
U
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⌠
⎮
⎮
⌡
d
=
But we have
u
U
0
y
δ
=
where we have used substitution

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