Problem 2.26
[Difficulty: 4]
Given:
Velocity field
Find:
Plot streamlines that are at origin at various times and pathlines that left origin at these times
Solution:
For streamlines
v
u
dy
dx
=
v
0
sin
ω
t
x
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
u
0
=
So, separating variables (t=const)
dy
v
0
sin
ω
t
x
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
u
0
dx
⋅
=
Integrating
y
v
0
cos
ω
t
x
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
ω
c
+
=
Using condition y = 0 when x = 0
y
v
0
cos
ω
t
x
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
⎣
⎤
⎥
⎦
cos
ω
t
⋅
()
−
⎡
⎢
⎣
⎤
⎥
⎦
⋅
ω
=
This gives streamlines y(x) at each time t
For particle paths, first find x(t)
dx
dt
u
=
u
0
=
Separating variables and integrating
dx
u
0
dt
⋅
=
o
r
xu
0
t
⋅
c
1
+
=
Using initial condition x = 0 at t =
τ
c
1
u
0
−
τ
⋅
=
0
t
τ
−
⋅
=
For y(t) we have
dy
dt
v
=
v
0
sin
ω
t
x
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
=
so
dy
dt
v
=
v
0
sin
ω
t
u
0
t
τ
−
⋅
u
0
−
⎡
⎢
⎣
⎤
⎥
⎦
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
=
and
dy
dt
v
=
v
0
sin
ωτ
⋅
⋅
=
Separating variables and integrating
dy
v
0
sin
⋅
⋅
dt
⋅
=
yv
0
sin
⋅
⋅
t
⋅
c
2
+
=
Using initial condition y = 0 at t =
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 Fall '07
 Lear
 Fluid Mechanics

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