Problem 6.112
[Difficulty: 4]
Given:
Complex function
Find:
Show it leads to velocity potential and stream function of irrotational incompressible flow; Show that df/dz leads
to u and v
Solution:
Basic equations: Irrotationality because
φ
exists
u
y
ψ
∂
∂
=
v
x
ψ
∂
∂
−
=
u
x
φ
∂
∂
−
=
v
y
φ
∂
∂
−
=
Incompressibility
x
u
∂
∂
y
v
∂
∂
+
0
=
Irrotationality
x
v
∂
∂
y
u
∂
∂
−
0
=
fz
() z
6
=
xi
y
⋅
+
()
6
=
Expanding
() x
6
15 x
4
⋅
y
2
⋅
−
15 x
2
⋅
y
4
⋅
+
y
6
−
i6xy
5
⋅
⋅
6x
5
y
⋅
⋅
+
20 x
3
⋅
y
3
⋅
−
⋅
+
=
We are thus to check the following
φ
xy
,
()x
6
15 x
4
⋅
y
2
⋅
−
15 x
2
⋅
y
4
⋅
+
y
6
−
=
ψ
,
6
x
⋅
y
5
⋅
5
⋅
y
⋅
+
20 x
3
⋅
y
3
⋅
−
−
=
uxy
,
x
φ
,
∂
∂
−
=
so
,
(
)
60 x
3
⋅
y
2
⋅
5
⋅
−
30 x
⋅
y
4
⋅
−
=
vxy
,
y
φ
,
∂
∂
−
=
so
,
(
)
30 x
4
⋅
y
⋅
60 x
2
⋅
y
3
⋅
−
6y
5
⋅
+
=
An alternative derivation of u and v is
,
y
ψ
,
∂
∂
=
,
(
)
60 x
3
⋅
y
2
⋅
5
⋅
−
30 x
⋅
y
4
⋅
−
=
,
x
ψ
,
∂
∂
−
=
,
(
)
30 x
4
⋅
y
⋅
60 x
2
⋅
y
3
⋅
−
5
⋅
+
=
Hence
x
,
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 Fall '07
 Lear
 Fluid Dynamics, Fluid Mechanics, Incompressible Flow, ∂x

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