1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

482 CHAPTER 11 • APPLI CATIONS OF HARMONIC F UNCTIONS

and the corresponding velocity function is

V(x, y) = F'(z) = 1 - (z)-^2 - ai(z)-^1.

We can express the complex potential in F = ¢> + i'I/! form:

F (z) = re'^8 + ~e-i^8 + ia (In r· + iO)

r

= (r + ; ) cos 0 - aO + i [ (r -~) sin 0 + a In r].

For the flow given by 'I/! = c, where c is a constant, we have

'l/J (rcosO, rsinO) = (r - ; ) sin 0 + aln1· = c (streamlines).

Setting r = 1 in this equation, we get 'I/! (cos 0, sin 0) = 0 for all 0 , so the unit

circle is a natural boundary curve for the flow.

Points at which the flow has zero velocity are called s tagnation points. To

find them we solve F' (z) = O; for the function in Equation (11-37) we have

1 - ~ + ai = 0. Multiplying through by z^2 and rearranging terms give

z z

z^2 + aiz - 1 = O. Now we invoke the quadratic equation to obtain

• ai ± V4-'i.i2

z =

2

(stagnation point(s)).

If 0 ~ lal < 2, then there are two stagnation points on the unit circle lzl = 1.

If a = 2, then there is one stagnation point on the unit circle. If !al > 2, then

the stagnation point lies outside the unit circle. We are mostly interested in the

case with two stagnation points. When a = 0, the two stagnation points are

z = ±1, which is the flow discussed in Example 11.25. The cases a = 1, a = v'3,

a= 2, and a = 2.2 are shown in Figure 11.65.

We are now ready to combine the preceding ideas. For illustrative purposes,

we consider a C1 circle with center co = -0.15 + 0.23i that passes through the

points z2 = 1 and z 4 = -1.3 and has radius r 0 = 0.23y'1372. We use the linear

transformation Z = S (z) = - 0.15 + 0.23i + r 0 z to map the flow with circulation

k = - 0.52p (or a= 0. 26 ) around izl = 1 onto the flow around the circle C 1 , as

shown in Figure 11.66.

Then we use the mapping w = J (Z) = Z + ~ to map this flow around the

Joukowski airfoil, as shown in Figure 11.67 and compare it to the flows shown in

Figures 11.63 and 11. 64. If the second transformation in the composition given

by w = J (z) =Sa (S2 (S 1 (z))) is modified to be S 2 (z) = zl.^925 , then t he image

of the flow shown in Figure 11.66 will be the flow around the modified airfoil