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Problem 7.1
[2]
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View Full Document Problem 7.2
[2]
Problem 7.3
[
2
]
Given:
Equation for beam
Find:
Dimensionless groups
Solution:
Denoting nondimensional quantities by an asterisk
L
x
x
L
I
I
t
t
L
y
y
L
A
A
=
=
=
=
=
*
*
*
*
*
4
2
ω
Hence
*
*
*
*
*
4
2
x
L
x
I
L
I
t
t
y
L
y
A
L
A
=
=
=
=
=
Substituting into the governing equation
0
*
*
*
1
*
*
*
4
4
4
4
2
2
2
2
=
∂
∂
+
∂
∂
x
y
LI
L
EL
t
y
A
L
L
ωρ
The final dimensionless equation is
0
*
*
*
*
*
*
4
4
2
2
2
2
=
∂
∂
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
∂
∂
x
y
I
L
E
t
y
A
The dimensionless group is
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
2
2
L
E
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View Full Document Problem 7.4
[2]
Problem 7.5
[4]
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View Full Document Problem 7.6
[
2
]
Given:
Equations for modeling atmospheric motion
Find:
Nondimensionalized equation; Dimensionless groups
Solution:
Recall that the total acceleration is
V
V
t
V
Dt
V
D
r
r
r
r
∇
⋅
+
∂
∂
=
Nondimensionalizing the velocity vector, pressure, angular velocity, spatial measure, and time, (using a typical velocity magnitude
V
and angular velocity magnitude
Ω
):
L
V
t
t
L
x
x
p
p
p
V
V
V
=
=
Ω
Ω
=
Ω
Δ
=
=
*
*
*
*
*
r
r
r
r
Hence
*
*
*
*
*
t
V
L
t
x
L
x
p
p
p
V
V
V
=
=
Ω
Ω
=
Ω
Δ
=
=
r
r
r
r
Substituting into the governing equation
*
1
*
*
2
*
*
*
*
*
p
L
p
V
V
V
V
L
V
V
t
V
L
V
V
∇
Δ
−
=
×
Ω
Ω
+
⋅∇
+
∂
∂
ρ
r
r
r
r
r
The final dimensionless equation is
*
*
2
*
*
*
*
*
2
p
V
p
V
V
L
V
V
t
V
∇
Δ
−
=
×
Ω
⎟
⎠
⎞
⎜
⎝
⎛ Ω
+
⋅∇
+
∂
∂
r
r
r
r
r
The dimensionless groups are
V
L
V
p
Ω
Δ
2
The second term on the left of the governing equation is the Coriolis force due to a rotating coordinate system.
This is a very
significant term in atmospheric studies, leading to such phenomena as geostrophic flow.
Problem 7.7
[
2
]
Given:
Equations Describing pipe flow
Find:
Nondimensionalized equation; Dimensionless groups
Solution:
Nondimensionalizing the velocity, pressure, spatial measures, and time:
L
V
t
t
L
r
r
L
x
x
p
p
p
V
u
u
=
=
=
Δ
=
=
*
*
*
*
*
Hence
*
*
*
*
*
t
V
L
t
r
D
r
x
L
x
p
p
p
u
V
u
=
=
=
Δ
=
=
Substituting into the governing equation
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
+
∂
∂
Δ
−
=
∂
∂
=
∂
∂
*
*
*
1
*
*
1
*
*
1
1
*
*
2
2
2
r
u
r
r
u
D
V
x
p
L
p
t
u
L
V
V
t
u
ν
ρ
The final dimensionless equation is
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
+
∂
∂
⎟
⎠
⎞
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛
+
∂
∂
Δ
−
=
∂
∂
*
*
*
1
*
*
*
*
*
*
2
2
2
r
u
r
r
u
D
L
V
D
x
p
V
p
t
u
The dimensionless groups are
D
L
V
D
V
p
2
Δ
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View Full Document Problem 7.8
[
2
]
Given:
Equation for unsteady, 2D compressible, inviscid flow
Find:
Dimensionless groups
Solution:
Denoting nondimensional quantities by an asterisk
0
0
0
0
0
*
*
*
*
*
*
*
c
L
L
c
t
t
c
c
c
c
v
v
c
u
u
L
y
y
L
x
x
ψ
=
=
=
=
=
=
=
Note that the stream function indicates volume flow rate/unit depth!
Hence
*
*
*
*
*
*
*
0
0
0
0
0
ψψ
c
L
c
t
L
t
c
c
c
v
c
v
u
c
u
y
L
y
x
L
x
=
=
=
=
=
=
=
Substituting into the governing equation
()
0
*
*
*
*
*
2
*
*
*
*
*
*
*
*
*
*
*
*
2
3
0
2
2
2
2
3
0
2
2
2
2
3
0
2
2
3
0
2
2
3
0
=
∂
∂
∂
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
∂
∂
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
∂
∂
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
∂
+
∂
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
∂
∂
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
y
x
v
u
L
c
y
c
v
L
c
x
c
u
L
c
t
v
u
L
c
t
L
c
The final dimensionless equation is
( )
0
*
*
*
*
*
2
*
*
*
*
*
*
*
*
*
*
*
*
2
2
2
2
2
2
2
2
2
2
2
2
2
=
∂
∂
∂
+
∂
∂
−
+
∂
∂
−
+
∂
+
∂
+
∂
∂
y
x
v
u
y
c
v
x
c
u
t
v
u
t
No dimensionless group is needed for this equation!
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This note was uploaded on 07/10/2011 for the course CHE 144 taught by Professor Tuzla during the Spring '11 term at Lehigh University .
 Spring '11
 TUZLA

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