Problem 2.11
[Difficulty: 3]
Given:
Flow field
Find:
Plot of velocity magnitude along axes, and y = x; Equation for streamlines
Solution:
On the x axis, y = 0, so
u
My
⋅
2
π
⋅
−
=
0
=
v
Mx
⋅
2
π
⋅
=
0
0.2
0.4
0.6
0.8
1
50
100
150
200
x (km)
v (m/s)
Plotting
The velocity is perpendicular to the axis and increases linearly with distance x.
This can also be plotted in Excel.
On the y axis, x = 0, so
u
⋅
2
π
⋅
−
=
v
⋅
2
π
⋅
=
0
=
0
0.2
0.4
0.6
0.8
1
200
−
150
−
100
−
50
−
y (km)
u (m/s)
Plotting
The velocity is perpendicular to the axis and increases linearly with distance y.
This can also be plotted in Excel.
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View Full DocumentOn the y = x
axis
u
My
⋅
2
π
⋅
−
=
Mx
⋅
2
π
⋅
−
=
v
⋅
2
π
⋅
=
The flow is perpendicular to line y = x:
Slope of line y =
x:
1
Slope of trajectory of
motion:
u
v
1
−
=
If we define the radial position:
rx
2
y
2
+
=
then along y =
x
2
x
2
+
=
2
x
⋅
=
Then the magnitude of the velocity along y = x is
Vu
2
v
2
+
=
M
2
π
⋅
x
2
x
2
+
⋅
=
M2
⋅
x
⋅
2
π
⋅
=
Mr
⋅
2
π
⋅
=
0
0.2
0.4
0.6
0.8
1
50
100
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 Fall '07
 Lear
 Fluid Mechanics, Derivative, M⋅

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