Problem 2.30
[Difficulty: 4]
Given:
Velocity field
Find:
Plot of pathline for t = 0 to 3 s for particle that started at point (1,2) at t = 0; compare to streakline through same
point at the instant t = 3
Solution:
Governing equations:
For pathlines
u
p
dx
dt
=
v
p
dy
dt
=
Following the discussion leading up to Eq. 2.10, we first find equations for the pathlines in form
x
p
t
( )
x t x
0
,
y
0
,
t
0
,
(
)
=
and
y
p
t
( )
y t x
0
,
y
0
,
t
0
,
(
)
=
x
st
t
0
(
)
x t x
0
,
y
0
,
t
0
,
(
)
=
and
y
st
t
0
(
)
y t x
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
=
a x
⋅
t
⋅
=
a
1
4
=
1
s
2
b
1
3
=
m
s
v
p
dy
dt
=
b
=
So, separating variables
dx
x
a t
⋅
dt
⋅
=
dy
b dt
⋅
=
Integrating
ln
x
x
0
⎛
⎜
⎝
⎞
⎠
a
2
t
2
t
0
2
−
⎛
⎝
⎞
⎠
⋅
=
y
y
0
−
b
t
t
0
−
(
)
⋅
=
y
y
0
b
t
t
0
−
(
)
⋅
+
=
x
x
0
e
a
2
t

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