1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 511

and

F'(z)=u+iv=w

The function F is called the complex velocity potential of the fl.ow, and

IF'(z)I = lwl is termed the speed of the fl.ow. From (7.29-5) we have

1

(x,y)

U(x,y)=('y) udx+vdy

(xo,Yo)

(7.29-6)

and

l

(x,y)

V(x,y) = (-y) u dy - v dx

(xo,yo)

(7.29-7)

The functions U(x, y) and V(x, y) are called the real velocity potential and
the stream function, respectively. The curves U(x, y) = c 1 are called equipo-
tentials, and the curves V(x,y) = c 2 are said to be the streamlines. The
two families of curves form an orthogonal system, except where F'(z) = O,
or w = O, i.e., at stagnation points.
Conversely, if F(z) = U + iV is an analytic function in a certain do-
main, the functions U(x, y) and V(x, y) give, respectively, the velocity
potential and stream function of a possible two-dimensional irrotational
and solenoidal fluid motion, and w = F'(z).
In steady motion the streamlines coincide with the trajectories of the
fluid elements since the differential equation of the streamlines is

Vx dx + Vy dy = 0 (7.29-8)

and the differential equation of the trajectories is

dx dy
u = v (7.29-9)
Equations (7.29-8) and (7.29-9) are identical, in view of Vx = -v and
Vy= u, as follows from (7.29-7). However, this is not necessarily the case
in a nonstationary fl.ow.

If we write the velocity field w = u +iv in the notation of vector analysis,

we have

w = ui + vj + Ok = ui + vj
where i,j, k are the unit vectors along the coordinate axes. Then the so-
called divergence and rotational of the two-dimensional vector w are given
by

div W = "\! • W = Ux + Vy

and