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23.4 Models of Plane Fluid Flow 785

2π

K r sin(t) K r cos(t)
udx + vdy = (−r sin(t)) − (r cos(t)) dt
2π r 2 2π r 2
γ 0
K 2π
= −dt =−K.
2π 0
This is the value of the circulation on any circle about the origin.
By a similar calculation,

−v dx + udy = 0.
γ
The origin is therefore neither a source nor a sink.
If we wish, we may restrict |z| > R to model flow having a solid cylinder about the origin as
a barrier with the flow swirling about the cylinder.

EXAMPLE 23.23
Interchange the roles of streamlines and equipotential curves in Example 23.22 by setting

f (z) = KLog(z)
with K a positive constant. Now
K
2
2
f (z) = ln(x + y ) + iKθ,
2
so
K 2 2
ϕ(x, y) = ln(x + y ) and ψ(x, y) = Kθ.
2
The equipotential curves are circles about the origin, and the streamlines are half-lines emanating
from the origin. The velocity of the flow is
K x y

f (z) = = K + iK = u + iv.
z x + y 2 x + y 2
2
2
If γ is a circle of radius r about the origin, then a straightforward calculation yields

udx + v dy = 0 and −v dx + udy = 2π K.
γ γ
The origin is a source of strength 2π K. This flow streams out from the origin, moving along
straight lines with velocity that decreases with distance from the source.

EXAMPLE 23.24
We will model flow around an elliptical barrier using a conformal mapping. From Example 23.22,
f (z)=(iK/2π)Log(z) for |z|> R models flow with circulation −K about a cylindrical barrier of
radius R centered at the origin. If the barrier is elliptical, we can combine this complex potential
with a mapping taking the circle |z|= R to an ellipse. To do this, consider the mapping
a 2
w = z +
z
in which a is a positive constant. This is a Joukowski transformation, and it is used in modeling
fluid flow around airplane wings because of the different images of the circle that occur by
making different choices for a.

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October 14, 2010 15:39 THM/NEIL Page-785 27410_23_ch23_p751-788