Chapter 9: Transport phenomena
43
and the definitions
v
x
=
∂ψ
/
∂
y
,
v
y
=

∂ψ
/
∂
x
holds:
Φ
AB
=
ψ
(
B
)

ψ
(
A
)
. In general holds:
∂
2
ψ
∂
x
2
+
∂
2
ψ
∂
y
2
=

ω
z
In polar coordinates holds:
v
r
=
1
r
∂ψ
∂θ
=
∂φ
∂
r
,
v
θ
=

∂ψ
∂
r
=
1
r
∂φ
∂θ
For source flows with power
Q
in
(
x, y
) = (0
,
0)
holds:
φ
=
Q
2
π
ln(
r
)
so that
v
r
=
Q/
2
π
r
,
v
θ
= 0
.
For a dipole of strength
Q
in
x
=
a
and strength

Q
in
x
=

a
follows from superposition:
φ
=

Qax/
2
π
r
2
where
Qa
is the dipole strength. For a vortex holds:
φ
=
Γ
θ
/
2
π
.
If an object is surrounded by an uniform main flow with
v
=
ve
x
and such a large Re that viscous effects are
limited to the boundary layer holds:
F
x
= 0
and
F
y
=

Γ
v
. The statement that
F
x
= 0
is d’Alembert’s
paradox and originates from the neglection of viscous effects. The lift
F
y
is also created by
η
because
Γ
= 0
due to viscous effects. Henxe rotating bodies also create a force perpendicular to their direction of motion: the
Magnus effect
.
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 Spring '10
 elder
 Aerodynamics, boundary layers, Ludwig Prandtl

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