Problem 2.15
[Difficulty: 2]
Given:
Velocity field
Find:
Proof that the parametric equations for particle motion are
x
p
c
1
e
At
⋅
⋅
=
and
y
p
c
2
e
2A
⋅
t
⋅
⋅
=
; pathline that was at
(2,2) at t = 0; compare to streamline through same point, and explain why they are similar or not.
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
⋅
=
v
p
dy
dt
=
⋅
y
⋅
=
So, separating variables
dx
x
Adt
⋅
=
dy
y
⋅
dt
⋅
=
Integrating
ln x
() A
t
⋅
C
1
+
=
ln y
() 2A
⋅
t
⋅
C
2
+
=
xe
⋅
C
1
+
=
e
C
1
e
⋅
⋅
=
c
1
e
⋅
⋅
=
ye
⋅
t
⋅
C
2
+
=
e
C
2
e
⋅
t
⋅
⋅
=
c
2
e
⋅
t
⋅
⋅
=
The pathlines are
xc
1
e
⋅
⋅
=
yc
2
e
⋅
t
⋅
⋅
=
Eliminating t
2
e
⋅
t
⋅
⋅
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 Fall '07
 Lear
 Fluid Mechanics

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