Problem 2.18
[Difficulty: 2]
Given:
Timevarying velocity field
Find:
Streamlines at t = 0 s; Streamline through (3,3); velocity vector; will streamlines change with time
Solution:
For streamlines
v
u
dy
dx
=
a y
⋅
2
cos
ω
t
⋅
(
)
+
(
)
⋅
a x
⋅
2
cos
ω
t
⋅
(
)
+
(
)
⋅
−
=
y
x
−
=
At t = 0 (actually all times!)
dy
dx
y
x
−
=
So, separating variables
dy
y
dx
x
−
=
Integrating
ln y
( )
ln x
( )
−
c
+
=
The solution is
y
C
x
=
which is the equation of a hyperbola.
For the streamline through point (3,3)
C
3
3
=
C
1
=
and
y
1
x
=
The streamlines will not change with time since dy/dx does not change with time.
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 Fall '07
 Lear
 Fluid Mechanics

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