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Problem 2.19
[Difficulty: 3]
Given:
Velocity field
Find:
Plot of pathline traced out by particle that passes through point (1,1) at t = 0; compare to streamlines through same
point at the instants t = 0, 1 and 2s
Solution:
Governing equations:
For pathlines
u
p
dx
dt
=
v
p
dy
dt
=
For streamlines
v
u
dy
dx
=
Assumption:
2D flow
Hence for pathlines
u
p
dx
dt
=
A1 B
t
⋅
+
()
⋅
=
A1
m
s
⋅
=
B1
1
s
⋅
=
v
p
dy
dt
=
Ct
⋅
y
⋅
=
C1
1
s
2
⋅
=
So, separating variables
dx
A 1
B t
⋅
+
⋅
dt
⋅
=
dy
y
⋅
dt
⋅
=
Integrating
xA
tB
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
C
1
+
=
ln y
1
2
C
⋅
t
2
⋅
C
2
+
=
ye
1
2
C
⋅
t
2
⋅
C
2
+
=
e
C
2
e
1
2
C
⋅
t
2
⋅
⋅
=
c
2
e
1
2
C
⋅
t
2
⋅
⋅
=
The pathlines are
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
C
1
+
=
yc
2
e
1
2
C
⋅
t
2
⋅
⋅
=
Using given data
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
1
+
=
1
2
C
⋅
t
2
⋅
=
For streamlines
v
u
dy
dx
=
Cy
⋅
t
⋅
t
⋅
+
⋅
=
So, separating variables
1B
t
⋅
+
dy
y
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 Fall '07
 Lear
 Fluid Mechanics

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