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23.4 Models of Plane Fluid Flow 783

since θ is real. This implies that the velocity has constant magnitude K.Insum, f models a
uniform flow with velocity of a constant magnitude K moving at an angle π −θ with the positive
real axis.

EXAMPLE 23.21
2
We will analyze the flow having complex potential f (z) = z . Since f (z) = 2z = 0 exactly when

z = 0, the origin is the only stagnation point. We will determine the trajectories of the flow.
2
2
With z = x + iy, f (z) = x − y + 2ixy,so
2
2
ϕ(x, y) = x − y and ψ(x, y) = 2xy.
2
2
Equipotential curves are hyperbolas x − y =k if k =0, and straight lines y =±x if k =0. These
are asymptotes of the hyperbolic equipotential curves. Streamlines are hyperbolas xy =c if c =0,
and the axes are x = 0 and y = 0if c = 0. Some streamlines and equipotential lines are shown in
Figure 23.34.
The velocity of the flow is f (z) = 2z. f models a nonuniform flow having velocity of

magnitude 2|z| at z. This flow moves along the hyperbolic streamlines with the axes acting as
barriers of the flow (think of sides of a container holding the fluid).
EXAMPLE 23.22
Consider the flow associated with the complex potential
iK
f (z) = Log(z),
2π
for z = 0, where Log(z) is the logarithm of z having argument between 0 and 2π. Thus,
1
2
2
Log(z) = ln(x + y ) + iθ,
2
x
2

1

y
−2 −1 0 1 2

−1

−2

FIGURE 23.34 Streamlines and equipotential lines in
Example 23.21.

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