1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

504 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

and obtain the equation x^2 + 2xycotc -y^2 = 1. If we express this equation in
the form

(

xcosc. c) (. c c)

2

• ysm

2

xsm

2

+ycos

2

. c c sin c
= sm-cos- = --
2 2 2

and use the rotation of axes

• c - c •. - c -c

x• = xcos

2

+ ysin

2

and y = -xsm

2

+ ycos

2

,

then the streamlines must satisfy the equation x•y• = si;c and are rectangular

hyperbolas with centers at the origin that pass through the points ± 1. Several

streamlines are indicated in Figure 11.95(b).

Let an ideal fluid flow in a domain in the z plane be affected by a source

located at the point zo. Then the flow at points z , which lie in a small neigh-

borhood of the point zo, is approximated by that of a source with the complex

potential

log (z - zo) + constant.

If w = S ( z) is a conformal mapping and w 0 = S ( zo), then S ( z) has a nonzero
derivative at zo and

w -wo = (z -zo) (S' (zo) + 11 (z)],

where 11 (z)-+ 0 as z -+ zo. Taking logarithms yields

log (w - wo) = log (z - zo) + Log [S' (zo) + 1J (z)].

Because S' (zo) :f 0, the term Log(S' (zo) + 11(z)] approaches the constant value
Log [S' (zo)] as z -+ zo. As log (z - zo) is the complex potential for a source
located at the point zo, the image of a source under a conformal mapping is a
source.
We can use the technique of conformal mapping to determine the fluid flow
in a domain D in the z plane that is produced by sources and sinks. If we can

construct a conformal mapping w = S (z) so that the image of sources, sinks, and

boundary curves for the flow in D are mapped onto sources, sinks, and boundary

curves in a domain G where the complex potential is known to be F 1 (w), then

the complex potential in Dis given by F2 (z) = F 1 (S(z)).

• EXAMPLE 11.32 Suppose that the lines x = ~'Tr are considered as walls of

a containing vessel for a fluid flow produced by a single source of unit strength