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Problem 2.29
[Difficulty: 4]
Given:
Velocity field
Find:
Plot of streakline for t = 0 to 3 s at point (1,1); compare to streamlines through same point at the instants t = 0, 1
and 2 s
Solution:
Governing equations:
For pathlines
u
p
dx
dt
=
v
p
dy
dt
=
For streamlines
v
u
dy
dx
=
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
,
=
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)
Assumption:
2D flow
For pathlines
u
p
dx
dt
=
ax
⋅
1b
t
⋅
+
⋅
=
a1
=
1
s
b
1
5
=
1
s
v
p
dy
dt
=
cy
⋅
=
c1
=
1
s
So, separating variables
dx
x
a1 b
t
⋅
+
⋅
dt
⋅
=
dy
y
cd
t
⋅
=
Integrating
ln
x
x
0
⎛
⎜
⎝
⎞
⎠
at t
0
−
b
t
2
t
0
2
−
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
=
ln
y
y
0
⎛
⎜
⎝
⎞
⎠
ct t
0
−
⋅
=
yy
0
e
ctt
0
−
⋅
⋅
=
xx
0
e
att
0
−
b
t
2
t
0
2
−
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
⋅
=
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View Full DocumentThe pathlines are
x
p
t
()
x
0
e
att
0
−
b
t
2
t
0
2
−
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
⋅
=
y
p
t
y
0
e
ctt
0
−
⋅
⋅
=
where x
0
, y
0
is the position of the particle at t = t
0
. Reinterpreting the results as streaklines:
The streaklines are then
x
st
t
0
x
0
e
0
−
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 Fall '07
 Lear
 Fluid Mechanics

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