Complex Analysis - Maths KU

444 Chapter 7 Conformal Mappings
In Problems 9 and 10, (a) find a linear fractional transformation of the domain
shown in color onto an annulus, and (b) use the mapping from (a) and a solution
similar to that in Example 1 to find the electrostatic potential φ(x, y) in the domain
subject to the given boundary conditions.
9. y
10.
–1
φ = 0
∇ φ = 0
2
1
φ = 10
Figure 7.71 Figure for Problem 9
2
x
–1
φ = 1
y
∇ φ = 0
2
φ = 0
Figure 7.72 Figure for Problem 10
In Problems 11 and 12, (a) find a conformal mapping of the domain shown in color
onto the domain used in Example 3, and (b) use the mapping from (a) and a solution
similar to that in Example 3 to find the steady-state temperature φ(x, y) in the
domain subject to the given boundary conditions.
11. y
12.
∇ φ = 0
2
x
φ = 0 –1 dφ
= 0 1 φ = 10
dn —
dφ
= 0
dn —
Figure 7.73 Figure for Problem 11
7.5.2 Fluid Flow
dφ
= 0
dn —
dφ
= 0
dn —
i
y
0
2
φ = 5
∇ φ = 0
2
x
φ = –10
Figure 7.74 Figure for Problem 12
In Problems 13–16, find the complex velocity potential Ω(z) for the flow of an ideal
fluid in the domain shown in color.
13. y
14.
y
π/4
Figure 7.75 Figure for Problem 13
x
Figure 7.76 Figure for Problem 14
x
x