1550251515-Classical_Complex_Analysis__Gonzalez_

Integration

y

vu

(^0) x
Fig. 7.31
is called the vorticity (at each point), and

C = k\i'^2 V

is the vorticity vector.

513

The fl.ow may also be derived from the stream function V, which always

exists regardless of whether the motion is irrotational or not. In fact, we

have

-2i ~~ = -i(V., + iVy) =Vy - iV., = u +iv= w (7.29-12)

Suppose that z 0 E D is such that u + iv is continuously differentiable in

some deleted neighborhood NHzo), and such that the circulation integral

j Wr ds = j u dx + v dy
c+ c+
has a nonzero constant value k for all closed C around z 0 described once
in the positive direction and such that C* C N6(z 0 ). Then the point z 0
is called a vortex and the number k is termed the strength (or intensity)
of the vortex.
Similarly, suppose that the flux integral

j w,, ds = j u dy - v dx

c+ c+

has a nonzero constant value m for all closed C around z 0 described once
in the positive direction and with graph contained in Nf;(z 0 ). Then the

point z 0 is called a source if m > 0 and a sink if m < 0. In either case the

number !ml is said to be the strength of the source or sink. The vortices