11.11 • SOURCES AND SINKS 5 0 5Figure 11.9 6 A source in the center of a strip.
located at the origin. T he conformal mapping w = S ( z) = sin z maps t he infinite
strip bounded by the lines x = ~1f onto the w plane slit along the boundary
rays u ~ -1, v = 0 and u 2:: 1, v = 0, and the image of the source at z 0 = 0 is a
source located at wo = 0. The complex potential
Fi (w) = logw
determines a fluid flow in thew plane past the boundary curves u ~ - 1, v = 0
and u 2:: 1, v = 0, which lie along streamlines of the flow. Therefore, the complex
potential for the fluid flow in the infinite strip in the z plane isF2 (z) = log (sin z).
Several streamlines for the flow are illustrated in Figure 11.96.
• EXAM PLE 11. 33 Suppose that the lines x = ~'Ir a.re considered as walls of
a containing vessel for the fluid flow produced by a single source of unit strength
1f
located at the point z 1 =
2
and a sink of unit stre ngth located a t the point- 1f
z2 =
2
. The conformal mapping w = S (z) = sin z maps the infinite strip
bounded by the lines x = ±; onto the w plane slit along the boundary rays
K 1 : u ~ -1, v = 0 and K2 : u 2:: 1, v = O. The image of the source at z 1 is
a source at w 1 = 1, and the image of the sink at z2 is a sink at w2 = - 1. The
potential
w-1
Fi (w) = log- -
w+ 1