Problem 6.101
[Difficulty: 3]
Given:
Data from Table 6.2
Find:
Stream function and velocity potential for a source in a corner; plot; velocity along one plane
Solution:
From Table 6.2, for a source at the origin
ψ
r
θ
,
(
)
q
2
π
⋅
θ
⋅
=
ϕ
r
θ
,
(
)
q
2
π
⋅
−
ln r
( )
⋅
=
Expressed in Cartesian coordinates
ψ
x y
,
(
)
q
2
π
⋅
atan
y
x
⎛
⎜
⎝
⎞
⎠
⋅
=
ϕ
x y
,
(
)
q
4
π
⋅
−
ln x
2
y
2
+
(
)
⋅
=
To build flow in a corner, we need image sources at three locations so that there is symmetry about both axes.
We need sources at
(
h
,
h
), (
h
,-
h
), (-
h
,
h
), and (-
h
,-
h
)
Hence the composite stream function and velocity potential are
ψ
x y
,
(
)
q
2
π
⋅
atan
y
h
−
x
h
−
⎛
⎜
⎝
⎞
⎠
atan
y
h
+
x
h
−
⎛
⎜
⎝
⎞
⎠
+
atan
y
h
+
x
h
+
⎛
⎜
⎝
⎞
⎠
+
atan
y
h
−
x
h
+
⎛
⎜
⎝
⎞
⎠
+
⎛
⎜
⎝
⎞
⎠
⋅
=
ϕ
x y
,
(
)
q
4
π
⋅
−
ln
x
h
−
(
)
2
y
h
−
(
)
2
+
⎡
⎣
⎤
⎦
x
h
−
(
)
2
y
h
+
(
)
2
+
⎡
⎣
⎤
⎦
⋅
⎡
⎣
⎤
⎦
⋅
q
4
π
⋅
x
h
+
(
)
2
y
h
+
(
)
2
+
⎡
⎣
⎤
⎦
⋅
x
h
+
(
)
2
y
h
−
(
)
2
+
⎡
⎣
⎤
⎦
⋅
−
=
By a similar reasoning the horizontal velocity is given by
u
q
x
h
−
(
)
⋅
2
π

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