1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

478 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

Figure 11.60

y

Image of a fluid flow under w = J (z) = z + !.

z

He showed that the image of a circle passing through z 1 = 1 and containing

the point z 2 = - 1 is mapped onto a. curve shaped like the cross section of an

airplane wing. We call this curve the J oukowski airfoil. If the streamlines for a

flow around the circle are known, then their images under the mapping w = J (z)

will be streamlines for a flow around the Joukowski airfoil, as shown in Figure
11.60.
The mapping w = J (z) is two-to-one, because J (z) = J (~),for z -f 0. The
region Jzl > 1 is mapped one-to-one onto the w plane slit a.Jong the portion of
the real axis -2 ~ u ~ 2. To visualize this mapping, we investigate the implicit
form, which we obtain by using the substitutions

1 z^2 -2z+ l

w-2 = z - 2+-= ----
z z
1 z^2 + 2z + 1
w+2 = z+2+-= ----
z z

(z-1)^2

z

(z + 1)

2

z

and

Forming the quotient of these two quantities results in the relationship

w-2- (z-1)

2

w + 2 z + l

The inverse of T ( w) = w -

2

2

is S 3 ( z) =

2

1

+

2

z. Therefore, if we use the

w + - z

notation S 1 (z) = z -

1

and 82 (z) = z^2 , we can express J (z) as the composition

z + l

of S1, 82, and 83:

w = J (z) = S3 (82 (Si (z))). (11-36)

We can easily show that w = J (z) = z + ~maps the four points z 1 = -i, z2 = 1,

z

zs = i, and z 4 = - 1 onto w 1 = 0, w2 = 2, w3 = 0, and w 4 = -2, respectively.

However, the composition functions in Equation (11-36) must be considered

in order to visualize the geometry involved. First, the bilinear transformation