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784 CHAPTER 23 Conformal Mappings and Applications

where θ is the argument of z lying in [0,2π). We therefore may write

iK 1 2 2 Kθ iK 2 2
f (z) = ln(x + y ) + iθ =− + ln(x + y ).
2π 2 2π 4π

Then
K K
2
2
ϕ(x, y) =− θ and ψ(x, y) = ln(x + y ).
2π 2π
Equipotential curves are graphs of θ = constant, which are half-lines from the origin making
angle θ with the positive real axis. Streamlines are graphs of ψ(x, y) = c, and these are circles
about the origin. These are trajectories of the fluid, which can be envisioned as moving in circular
paths about the origin. Some streamlines and equipotential curves are shown in Figure 23.35.

Compute f (z) = (iK/2πz) if z = 0. On the circle |z|=r, the magnitude of the velocity is
K 1 K

| f (z)|= = .
2π |z| 2πr
This velocity increases as r → 0, so the fluid swirls about the origin with increasing velocity
toward the center (origin). The origin is a vortex of this flow.
To calculate the circulation of the flow about the origin, write

iK 1 iK z K y iK x

f (z) =− =− = − = u + iv.
2π z 2π |z| 2 2π x + y 2 2π x + y 2
2
2
If γ is the circle of radius r about the origin, then on γ ,wehave x =r cos(t) and y =r sin(t) for
0 ≤ t ≤ 2π,so

y
2

1

x
−2 −1 0 1 2

−1

−2

FIGURE 23.35 Streamlines and equipotential lines in
Example 23.22.

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October 14, 2010 15:39 THM/NEIL Page-784 27410_23_ch23_p751-788