Complex Analysis - Maths KU

290 Chapter 5 Integration in the Complex Plane
y
F F
Figure 5.49 Uniform flow
–2
–1
2
1
–1
–2
y
Figure 5.50 Positive circulation and
zero net flux
C
1
2
x
x
EXAMPLE 3 Complex Potentials
(a) The analytic function Ω(z) = 1
2 kz2 , k > 0, is a complex potential for
the flow in Example 2. By (16), the derivative g(z) =Ω ′ (z) =kz is an
analytic function. By (17), the conjugate f(z) =Ω ′ (z) orf(z) =k¯z =
kx − iky is the complex representation of the velocity vector field F of
the flow of an ideal fluid in some domain D of the plane. The complex
potential of F is Ω(z). Since Ω(z) = 1
2 k(x2 − y 2 +2xyi), we see that the
streamlines of the flow are xy = c2.
(b) The complex function Ω(z) =Az, A > 0, is a complex potential for a
very simple but important type of flow. From
Ω ′ (z) =A and Ω ′ (z) =A,
we see that Ω(z) is the complex potential of vector field F whose complex
representation is f(z) =A. Because the speed |f| = A is constant at every
point, we say that the velocity field F(x, y) =Ai is a uniform flow. In
other words, in a domain D such as the upper half-plane, a particle in the
fluid moves with a constant speed. From Ω(z) =Az = Ax + iAy, we see
that a path of a moving particle, a streamline for the flow, is a horizontal
line fromthe family defined by y = c2. Notice that the boundary of the
domain D, y = 0, is itself a streamline. See Figure 5.49.
Circulation and Net Flux Given a simple closed curve C oriented
counterclockwise in the plane and a complex function f that represents the
velocity field of a planar fluid flow, we can ask the following two questions:
(i) To what degree does the fluid tend to flow around the curve C?
(ii) What is the net difference between the rates at which fluid enters and
leaves the region bounded by the curve C?
The quantities considered in questions (i) and (ii) are called the circulation
around C and the net flux across C, respectively. A precise definition of
circulation and flux depends on the use of a contour integral involving the
complex representation f and will be given shortly. In the meantime, we
can decide whether the circulation or net flux is positive, negative, or 0 by
graphing the velocity vector field f of the flow. As with arguments of complex
numbers, we consider the counterclockwise direction of a flow as the “positive”
direction. Thus, a flow will have a positive circulation around C if the fluid
tends to flow counterclockwise around C. Similarly, a negative circulation
means the fluid tends to flow clockwise around C, and a 0 circulation means
that the flow is perpendicular to C. For example, in Figure 5.50, the circulation
is positive since the fluid tends to flow counterclockwise around C, whereas
the circulation in Figure 5.51 is 0 since the flow is perpendicular to the curve
C. In a similar manner, we consider a positive net flux to mean that fluid