Complex Analysis - Maths KU

442 Chapter 7 Conformal Mappings
4
3
2
1
y
1 2 3 4 x
Figure 7.61 Figure for Example 8
y
(a) Flow around the unit circle
y
(b) Flow around a circle containing
z1 = –1 and passing through z2 = 1
y
(c) Flow around an airfoil
Figure 7.62 Flow around a Joukowski
airfoil
x
x
x
we obtain a complex potential function for a flow in D byreplacing the symbol
w with z 4 in (13). This yields
Ω(z) =Ln � z 4 − 1 � − Ln � z 4 − 81 � . (14)
Streamlines of this flow are given by ψ(x, y) =c2, or
See Figure 7.61.
Remarks
Arg � z 4 − 1 � − Arg � z 4 − 81 � = c2.
The complex mapping w = J(z) =z + k 2 /z is called the Joukowski
transformation. Under this mapping, a circle in the z-plane that contains
the point z1 = −1 and passes through the point z2 = 1 is mapped
onto a curve in the w-plane that resembles the cross-section of an airplane
wing. See Figure 7.62(c). The image curve is called a Joukowski airfoil,
and the air flow around this curve can be determined using techniques
from this section. We begin with the flow shown in Figure 7.62(a) given
by Ω(z) =z +1/z around the unit circle |z| = 1. Using an appropriate
linear mapping, we can adjust this flow to be one around a circle containing
the point z1 = −1 and passing through the point z2 = 1. See Figure
7.62(b). The Joukowski transformation is then used to “transform” this
flow to one around the airfoil as shown in Figure 7.62(c).
EXERCISES 7.5 Answers to selected odd-numbered problems begin on page ANS-23.
7.5.1 Boundary-Value Problems
In Problems 1–6, (a) find a conformal mapping of the domain shown in color onto
the upper half-plane, and (b) use the mapping from (a) and the solution (7) in
Section 7.4 to find the steady-state temperature φ(x, y) in the domain subject to
the given boundary conditions.
1. y
2.
y
φ = 0
∇ φ = 0
2
i
φ = 1
φ = 2 1 φ = 0
Figure 7.63 Figure for Problem 1
x
φ = 0
i
∇ φ = 0
2
φ = 2
φ = 1 –1 1 φ = –1
Figure 7.64 Figure for Problem 2
x