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Problem 2.22
[Difficulty: 3]
Given:
Velocity field
Find:
Plot of pathline of particle for t = 0 to 1.5 s that was at point (1,1) at t = 0; compare to streamlines through same
point at the instants t = 0, 1 and 1.5 s
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
=
ax
=
a2
=
1
s
v
p
dy
dt
=
by
⋅
1c
t
⋅
+
()
⋅
=
b2
=
1
s
2
c
0.4
=
1
s
So, separating variables
dx
x
ad
t
⋅
=
dy
b y
⋅
t
⋅
+
⋅
dt
⋅
=
dy
y
b1 c
t
⋅
+
⋅
dt
⋅
=
Integrating
ln
x
x
0
⎛
⎜
⎝
⎞
⎠
at
⋅
=
x
0
1
=
m
ln
y
y
0
⎛
⎜
⎝
⎞
⎠
bt
1
2
c
⋅
t
2
⋅
+
⎛
⎝
⎞
⎠
⋅
=
y
0
1
=
m
Hence
xt
x
0
e
⋅
⋅
=
yt
e
1
2
c
⋅
t
2
⋅
+
⎛
⎜
⎝
⎞
⎠
⋅
=
Using given data
e
2t
⋅
=
e
⋅
0.4 t
2
⋅
+
=
For streamlines
v
u
dy
dx
=
⋅
t
⋅
+
⋅
ax
⋅
=
So, separating variables
dy
y
t
⋅
+
⋅
⋅
dx
⋅
=
which we can integrate for any given t (t is treated as a constant)
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 Fall '07
 Lear
 Fluid Mechanics

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