1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

500 CHAPTER 11 • APPLICATIONS OF HARMONIC FUNCTIONS

(a) A source al lhe origin. (b) A sink at the origin.

Figure 11.93 Sources and sinks for an ideal fluid.

simple source. A source at the origin can be considered as a line perpendicular

to the z plane along which fluid is being emitted. If the rate of emission of

volume of fluid per unit length is 211'm, then the origin is said to be a source of

strength m, the complex potential for the flow is

F(z) = mlogz,

and the velocity Vat the point (x, y) is given by

-- m

V (x, y) = F' (z) = -=·

z
For fluid flows, a sink is a negative source and is a point of inward radial flow at
which the fluid is considered to be absorbed or annihilated. Sources and sinks
for flows are illustrated in Figure 11.93.

11.11.1 Source: A C harged Line

In the case of electrostatics, a source will correspond to a uniformly charged line
perpendicular to the z plane at the point zo. We will show that if the line L is
located at zo = 0 and carries a charge density of § coulombs per unit length,
then the magnitude of the electrical field is IE (x, y)I = q Hence Eis
.,/x2 + y2
given by
qz q

E(x,y) = - 2 = =•

lzl z

and the complex potential is

F(z)=-qlogz and E(x,y)=-F'(z).

(11-41)

A sink for electrostatics is a negatively charged line perpendicular to the z plane.

The electric field for electrostatic problems corresponds to the velocity field for