1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

506 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

Figure 11.97 A source and a sink on the edges of a strip.

determines a fluid flow in the w plane past the boundary curves K 1 and K2,

which lie along streamlines of the flow. Therefore, the complex potential for the

fluid flow in the infinite strip in the z plane is

sin z - 1

F2(z)=log..

smz+ 1

Several streamlines for the flow are illustrated in Figure 11.97.

We can use the technique of transformation of a source to determine the

effluence from a channel extending from infinity. In this case, we construct a

conformal mapping w = S (z) from the upper ha.If-plane Im (z) > 0 so that

the single source located at zo = 0 is mapped to the point wo at infinity that

lies along the channel. The streamlines emanating from zo = 0 in the upper

half-plane are mapped onto streamlines issuing from the channel.

• EXAMPLE 11 .3 4 Consider the conformal mapping

2 1 2 1

w = S (z) = - (z^2 - 1)^2 + - Arcsin- ,

1f 1f z

which maps the upper half-plane lm(z) > 0 onto the domain consisting of the

upper half-plane Im ( w) > 0 joined to the channel - 1 $ u $ 1, v $ O. The point

zo = 0 is mapped onto the point w 0 = - ioo along the channel. Images of the

rays r > 0, (} = a: are streamlines issuing from the channel as indicated in Figure

11.98.