Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

is described by the complex potential

f(z)=kln(z−z 0 ),

wherekis the strength of the source. A sink is similarly represented, but withk

replaced by−k. Other simple examples are as follows.

(i) The flow of a fluid at a constant speedV 0 and at an angleαto thex-axis

is described byf(z)=V 0 (expiα)z.

(ii) Vortex flow, in which fluid flows azimuthally in an anticlockwise direction

around some pointz 0 , the speed of the flow being inversely proportional

to the distance fromz 0 , is described byf(z)=−ikln(z−z 0 ), wherekis

the strength of the vortex. For a clockwise vortexkis replaced by−k.

Verify that the complex potential

f(z)=V 0

(

z+

a^2

z

)

is appropriate to a circular cylinder of radiusaplaced so that it is perpendicular to a

uniform fluid flow of speedV 0 parallel to thex-axis.

Firstly, sincef(z)isanalyticexceptatz= 0, both its real and imaginary parts satisfy
Laplace’s equation in the region exterior to the cylinder. Alsof(z)→V 0 zasz→∞,so
that Ref(z)→V 0 x, which is appropriate to a uniform flow of speedV 0 in thex-direction
far from the cylinder.
Writingz=rexpiθand using de Moivre’s theorem we have

f(z)=V 0

[

rexpiθ+

a^2

r

exp(−iθ)

]

=V 0

(

r+

a^2

r

)

cosθ+iV 0

(

r−

a^2

r

)

sinθ.

Thus we see that the streamlines of the flow described byf(z) are given by

ψ=V 0

(

r−

a^2

r

)

sinθ=constant.

In particular,ψ=0onr=a, independently of the value ofθ,andsor=amust be a
streamline. Since there can be no flow of fluid across streamlines,r=amust correspond
to a boundary along which the fluid flows tangentially. Thusf(z) is a solution of Laplace’s
equation that satisfies all the physical boundary conditions of the problem, and so, by the
uniqueness theorem, it is the appropriate complex potential.

By a similar argument, the complex potentialf(z)=−E(z−a^2 /z) (note the

minus signs) is appropriate to a conducting circular cylinder of radiusaplaced

perpendicular to a uniform electric fieldEin thex-direction.

The real and imaginary parts of a complex potentialf=φ+iψhave another

interesting relationship in the context of Laplace’s equation in electrostatics or

fluid mechanics. Let us chooseφas the conventional potential, so thatψrepresents

the stream function (or electric field, depending on the application), and consider