Problem 2.20
[Difficulty: 3]
Given:
Velocity field
Find:
Plot of pathline traced out by particle that passes through point (1,1) at t = 0; compare to streamlines through
same point at the instants t = 0, 1 and 2s
Solution:
Governing equations:
For pathlines
u
p
dx
dt
=
v
p
dy
dt
=
For streamlines
v
u
dy
dx
=
Assumption:
2D flow
Hence for pathlines
u
p
dx
dt
=
B x
⋅
1
A t
⋅
+
(
)
⋅
=
A
0.5
1
s
⋅
=
B
1
1
s
⋅
=
v
p
dy
dt
=
C y
⋅
=
C
1
1
s
⋅
=
So, separating variables
dx
x
B
1
A t
⋅
+
(
)
⋅
dt
⋅
=
dy
y
C dt
⋅
=
Integrating
ln x
( )
B
t
A
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
C
1
+
=
ln y
( )
C t
⋅
C
2
+
=
y
e
C t
⋅
C
2
+
=
e
C
2
e
C t
⋅
⋅
=
c
2
e
C t
⋅
⋅
=
x
e
B
t A
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
C
1
+
=
e
C
1
e
B
t A
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
⋅
=
c
1
e
B
t A
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
⋅
=
The pathlines are
x
c
1
e
B
t A
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
⋅
=
y
c
2
e
C t
⋅
⋅
=
Using given data
x
e
B
t A
t
2
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
=
y
e
C t
⋅
=
For streamlines

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