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786 CHAPTER 23 Conformal Mappings and Applications

2
Let z = x + iy, and w = X + iY. The circle x + y = R is mapped to the ellipse
2
2
X 2 Y 2
+ = R 2
1 + (a/R) 2 1 − (a/R) 2
if a = R.If a = R, the circle maps to the interval [−2a,2a] on the real axis.
Solve for z in the Joukowski transformation to get
√
2
w + w − 4a 2
z = .
2
Compose this mapping with the complex potential function for the circular barrier to get
√

iK w + w − 4a 2
2
F(w) = f (z(w)) = Log .
2π 2
This is the complex potential for flow in the w-plane about an elliptical barrier if R >a and about
the flat plate −2a ≤ X ≤ 2a,Y = 0if R = a.
We conclude this section with an application of complex integration to fluid flow. Suppose f
is a complex potential for a flow about a barrier whose boundary is a closed path γ .Let Ai + Bj
be the thrust of the fluid outside the barrier. Blasius’s theorem states that
iρ
2

A + iB = ( f (z)) dz,
2 γ
in which ρ is the constant density of the fluid. Furthermore, the moment of the thrust about the
origin is

ρ
2

Re − z( f (z)) dz .
2 γ
SECTION 23.4 PROBLEMS

In each of Problems 1 through 4, analyze the flow hav- with K as a nonzero real numbers and a and b as
ing the given complex potential. Sketch some equipotential distinct complex numbers. Sketch some equipotential
curves and streamlines and determine the velocity, any curves and streamlines and determine any stagnation
stagnation points, and whether the flow has any sources points, sources, and sinks.
or sinks.
7. Let
1
1. f (z) = az with a a nonzero complex number. f (z) = K z +
z
2. f (z) = z 3
with K as a nonzero real number. Sketch some equipo-
3. f (z) = cos(z) tential curves and streamlines. Show that f models
4. f (z) = z + iz 2 flow around the upper half of the unit circle.
8. Let
5. f (z) = KLog(z − z 0 ) with K as a nonzero real con-
m − ik z − a
stant and z 0 as a complex number. Show that z 0 f (z) = Log
is a source if K > 0 and a sink if K < 0. Sketch 2π z − b
some equipotential curves and streamlines. Log(z) is with m and k as nonzero, real numbers and a and b as
defined here to be the branch of log(z) obtained by distinct, complex numbers. Show that this flow has a
restricting −π ≤ arg (z)<π. source or sink of strength m and a vortex of strength
k at both a and b. A point combining properties of a
6. Let
source or sink and a vortex is called a spiral vortex.

z − a Sketch some equipotential curves and streamlines for
f (z) = KLog this flow.
z − b

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