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780 CHAPTER 23 Conformal Mappings and Applications

a flow is called plane-parallel. This velocity vector is also assumed to be independent of time, in
which case we say that the flow is stationary. Write
V(x, y) = u(x, y)i + v(x, y)j.
Since we identify vectors in the plane with complex numbers, we will write this as

V (z) = V (x + iy) = u(x, y) + iv(x, y).
Given V (z), think of the complex plane as divided into two sets. The first is the domain D
on which V is defined. The complement of D consists of the points not in D. Think of this
complement as comprising channels or barriers confining the fluid to D. This enables us to
model fluid flow through a variety of configurations and around barriers of various shapes.
Suppose γ is a closed path in D. Write γ(s)=x(s)+iy(s), using arc length s as a parameter.

Then x (s)i + y (s)j is the unit tangent vector to γ . Furthermore,

dx dy

(ui + vj) · i + j ds = udx + vdy.
ds ds
Since udx + vdy is the dot product of the velocity with the unit tangent along the trajectory γ ,
we interpret

udx + v dy
γ
as a measure of the velocity of the fluid along γ . This integral is the circulation of the fluid
around γ .
The vector −y (s)i + x (s)j is a unit normal vector to γ (Figure 23.32). Then

dy dx
− (ui + vj) · i + j ds = −v dx + udy
ds ds
γ γ
is the negative of the integral of the normal component of the velocity along γ . When this integral
is not zero, it is called the flux of the fluid across the path. This gives a measure of fluid flow across
γ out of the region enclosed by γ . When this flux is zero for every closed path in the domain of
the fluid, we call the fluid solenoidal.
A point z 0 = x 0 + iy 0 is a vortex of the fluid if the circulation has a nonzero value k that is
the same for every closed path about z 0 in the interior of some annulus 0 < |z − z 0 | < r. We call
|k| the strength of the vortex.If k > 0the vortex is a source, and if k < 0itisa sink.
The following result is the key to using complex functions to analyze fluid flow.

y
T
γ

N

x

FIGURE 23.32 Unit tangent and
normal vectors to C.

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October 14, 2010 15:39 THM/NEIL Page-780 27410_23_ch23_p751-788