Elementary Planar Irrotational Flows in Complex Variables
Author: John M. Cimbala, Penn State University
Latest revision: 17 October 2007
Note: Consider steady, incompressible, irrotational, Newtonian fluid flow in which gravity is neglected. The flow
is assumed to be two-dimensional in the x-y or r-
θ
plane.
Summary of the Equations
• Complex potential:
(
)
(
)
() , ,wz x
y
ix
y
φψ
=+ or
(
)
(
)
() , ,wz r i r
φ
θψθ
=+ , where i
zxiyre
θ
=+ = ,
u
x
y
φ
ψ
∂∂
==
∂∂
, vyx
φ
ψ
∂∂
==−
∂∂
, 1
r
urr
φ
ψ
θ
∂∂
==
∂∂
, and 1
urr
θ
φ
ψ
θ
∂
∂
==−
∂
∂.
• Complex velocity:
()
i
(,) (,) (,) (,
ir
dw uxy ivxy qe u r iu r e
dz
α
θ
θ
θθ
−
−
=− == − where 22
quv=+
, 1
tan v
u
α
−⎛⎞
=⎜⎟
⎝⎠
.
Elementary Planar Irrotational Flows
x
ψ
= 0
ψ
1
y
U
ψ
2
-
ψ
2
-
ψ
1
ψ
3
φ
= 0
φ
1
φ
2 -
φ
2 -
φ
1
a. Uniform stream in the x-direction:
0
uU
v
=
=, ()w z Uz Ux iUy==+ , dw U
dz =, Ux
φ
=
, Uy
ψ
=
x
ψ
= 0
ψ
1
y
U
ψ
2
-
ψ
2
-
ψ
1
φ
= 0
φ
1
φ
2
-
φ
2 -
φ
1
α
b. Uniform stream in an
arbitrary direction:
cos
sin
uU
vU
α
=
α
=, () i
wz Uze
α
−
=,
()
cos sin i
dw UiU
dz e
α
αα
−
=−=
,
()
cos sinUx y
φ
αα
=+
,
(
)
cos sinUy x
ψ
αα
=−
.
c. Line source at the origin:
2
0
r
m
ur
u
θ
π
=
=
,
()
() ln ln
22
i
mm
wz z re
θ
ππ
== , 22
i
dw m m e
dz z r
θ
ππ
−
== ,
ln
2
mr
φπ
=, 2
m
ψ
θ
π
=.
d. Line vortex at the origin:
0
2
r
u
ur
θ
π
=
Γ
=,
θ
x
y
r
ψ
1
ψ
2
ψ
3
ψ
4
φ
1
φ
2
ψ
5
ψ
6
ψ
7
ψ
8
φ
3
m
θ
x
y
r
φ
1
φ
2
φ
3
φ
4
ψ
3
ψ
2
φ
5
φ
6
φ
7
φ
8
ψ
1
Γ
()
() ln ln
22
i
wz i z i re
θ
ππ
ΓΓ
=− =− , 22
i
dw ii
dz z r e
θ
ππ
−
ΓΓ
=− =− ,
2
φ
θ
π
Γ
=, ln
2r
ψπ
Γ
=− .