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Problem 2.27
[Difficulty: 5]
Given:
Velocity field
Find:
Plot streakline for first second of flow
Solution:
Following the discussion leading up to Eq. 2.10, we first find equations for the pathlines in form
x
p
t
()
xtx
0
,
y
0
,
t
0
,
=
and
y
p
t
ytx
0
,
y
0
,
t
0
,
=
where x
0
, y
0
is the position of the particle at t = t
0
, and reinterprete the results as streaklines
x
st
t
0
0
,
y
0
,
t
0
,
=
and
y
st
t
0
0
,
y
0
,
t
0
,
=
which gives the streakline at t, where x
0
, y
0
is the point at which dye is released (t
0
is varied from 0 to t)
For particle paths, first find x(t)
dx
dt
u
=
u
0
=
Separating variables and integrating
dx
u
0
dt
⋅
=
o
r
xx
0
u
0
tt
0
−
⋅
+
=
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
x
0
u
0
0
−
⋅
+
u
0
−
⎡
⎢
⎣
⎤
⎥
⎦
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
=
and
dy
dt
v
=
v
0
sin
ω
t
0
x
0
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
⎣
⎤
⎥
⎦
⋅
=
Separating variables and integrating
dy
v
0
sin
ω
t
0
x
0
u
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⎡
⎢
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 Fall '07
 Lear
 Fluid Mechanics

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