1550251515-Classical_Complex_Analysis__Gonzalez_

516 Chapter^7

On the other hand, the complex potential

m z-a

F( z) = 2 7r log z _ b

with m > 0 corresponds to a fl.ow with a source at a and a sink at b,

both of strength m. The fl.ow pattern for this case is shown in Fig. 7.33.

The streamlines are arcs of circles joining the points a and b, and the

equipotential lines are the circles orthogonal to those arcs.

Another interpretation of the logarithmic singularity of the complex

potential can be obtained by interchanging the roles of the stream and

equipotential lines. This is accomplished by multiplying the complex

potential by i. Thus by writing

mi

F(z) = - logz

27r

we have the velocity potential

-m

U =

2

7!" argz

and for the stream function

m

V= -lnlzl

27r

so that the equipotentials are now radial lines from the origin while the

streamlines are concentric circles with centers at the origin. As z --t 0 the

velocity becomes infinitely large, and the circulation integral has the value

-m. Hence the origin is in this case a vortex with strength -m.

Fig. 7.33