D Some simple flows and their potentials
D.1 Two dimensional flows
We will give the answer either in Cartesians or polars, depending on which is more con
venient.
Reminder:
(i)
in Cartesians,
u
= (
u,v,
0) and
u
=
∂φ
∂x
=
∂ψ
∂y
,
v
=
∂φ
∂y
=

∂ψ
∂x
where
φ
is the
velocity potential and
ψ
is the streamfunction.
(ii)
In polars,
u
= (
u
r
,u
θ
,
0) and
u
r
=
∂φ
∂r
=
1
r
∂ψ
∂θ
,
u
θ
=
1
r
∂φ
∂θ
=

∂ψ
∂r
.
Type of flow
flow field
u
potential
φ
streamfunction
ψ
complex pot. w
Uniform stream par
allel to
x
axis
U
ˆ
x
Ux
Uy
Uz
Uniform
stream
at
angle
α
to
x
axis
U
cos
α
ˆ
x
+
U
sin
α
ˆ
y
Ur
cos(
θ

α
)
Ur
sin(
θ

α
)
Uze

iα
Stagnation point flow
at origin
u
=
kx
v
=

ky
k
2
(
x
2

y
2
)
kxy
k
2
z
2
Line source, strength
m
, at
r
= 0
m
ˆ
r
2
πr
m
2
π
log
r
mθ
2
π
m
2
π
ln
z
Line vortex, circula
tion Γ, at
r
= 0
Γ
ˆ
θ
2
πr
Γ
θ
2
π

Γ
2
π
log
r

i
Γ
2
π
ln
z
Horizontal
dipole,
strength
μ
, at
r
= 0

μ
2
πr
2
parenleftBig
ˆ
z

2
z
r
ˆ
r
parenrightBig

μ
cos
θ
2
πr
μ
sin
θ
2
πr

μ
2
πz
D.2 Axisymmetric flows
We use cylindrical or spherical polars, whichever is more convenient. In cylindrical coor
dinates, (
r,θ,z
),
u
= (
u
r
,u
θ
,u
z
); in terms of the potential
φ
(
r,z
) or the Stokes stream
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 Fall '11
 Eggers
 Fluid Dynamics, Coordinate system, Polar coordinate system, streamfunction ψ

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