Problem 2.31
[Difficulty: 4]
Given:
2D velocity field
Find:
Streamlines passing through (6,6); Coordinates of particle starting at (1,4); that pathlines, streamlines and
streaklines coincide
Solution:
For streamlines
v
u
dy
dx
=
b
ay
2
⋅
=
or
y
2
⋅
⌠
⎮
⎮
⌡
dx
b
⌠
⎮
⌡
d
=
Integrating
3
⋅
3
bx
⋅
c
+
=
For the streamline through point (6,6)
c6
0
=
and
y
3
6x
⋅
180
+
=
For particle that passed through (1,4) at t = 0
u
dx
dt
=
2
⋅
=
x
1
⌠
⎮
⌡
x
0
−
=
t
2
⋅
⌠
⎮
⎮
⌡
d
=
We need y(t)
v
dy
dt
=
b
=
y
1
⌠
⎮
⌡
dt
b
⌠
⎮
⌡
d
=
yy
0
bt
⋅
+
=
y
0
2t
⋅
+
=
Then
xx
0
−
0
t
t
0
⋅
+
()
2
⋅
⌠
⎮
⌡
d
=
0
0
2
t
⋅
by
0
⋅
t
2
⋅
+
b
2
t
3
⋅
3
+
⎛
⎜
⎝
⎞
⎠
⋅
+
=
Hence, with
x
0
1
=
y
0
4
=
x
1
16 t
⋅
+
8t
2
⋅
+
4
3
t
3
⋅
+
=
At
t = 1 s
x
26.3 m
⋅
=
y42
t
⋅
+
=
y6
m
⋅
=
For particle that passed through (3,0) at t = 1
y
1
⌠
⎮
⌡
b
⌠
⎮
⌡
d
=
0
bt t
0
−
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 Fall '07
 Lear
 Fluid Mechanics

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