Center:
.
1
Radius:
.
5
Center:
.
2 + i
Radius: 1
Center: 1 + i
Radius: 1
Center:
−
2 + 3 i
Radius: 3
√
2
≈
4
.
2426
Center:
.
2 + i
Radius: 1
.
2806
Center:
.
1 +
.
3 i
Radius:
.
9487
Center:
.
1 +
.
1 i
Radius: 1
.
1045
Center:
−
.
2 +
.
1 i
Radius: 1
.
2042
Figure 7.21.
Airfoils Obtained from Circles via the Joukowski Map.
rest of the
ζ
plane, as do the images of the (nonzero) points inside the unit circle. Indeed,
if we solve (7.67) for
z
=
ζ
±
r
ζ
2
−
1
,
(7
.
68)
we see that every
ζ
except
±
1 comes from two diFerent points
z
; for
ζ
not on the critical
line segment [
−
1
,
1], one point lies inside and and one lies outside the unit circle, whereas
if
−
1
< ζ <
1, the points lie on the unit circle and on a common vertical line. Therefore,
(7.67) de±nes a onetoone conformal map from the exterior of the unit circle
b

z

>
1
B
onto the exterior of the unit line segment
C
\
[
−
1
,
1].
Under the Joukowski map, the concentric circles

z

=
r
n
= 1 are mapped to ellipses
with foci at
±
1 in the
ζ
plane; see ²igure 7.20. The eFect on circles not centered at the
origin is quite interesting. The image curves take on a wide variety of shapes; several
examples are plotted in ²igure 7.21. If the circle passes through the singular point
z
= 1,
then its image is no longer smooth, but has a cusp at
ζ
= 1; this happens in the last 6
of the ±gures. Some of the image curves have the shape of the crosssection through an
airplane wing or
airfoil
. Later, we will see how to construct the physical ³uid ³ow around
such an airfoil, a result that was a critical step in early aircraft design.
Composition and the Riemann Mapping Theorem
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 Fall '10
 Olver
 Differential Equations, Equations, Partial Differential Equations, Conformal map, Unit disk, conformal maps, right half plane

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