Complex Analysis - Maths KU

7.5 Applications 447
28. In this problem you will construct the flow of an ideal fluid around a plate
shown Figure 7.89.
(a) Use a linear mapping and the velocity potential from Example 5 to show
that the velocity potential of an ideal fluid in the domain shown in Figure
7.88 is given by
Ω(z) = z eiα
+
eiα z .
(b) The domain outside of the unit circle shown in Figure 7.88 is mapped onto
the complex plane excluding the line segment y =0, −2 ≤ x ≤ 2, shown in
Figure 7.89 by the conformal mapping
w = z + � z 2 − 4 �1/2 .
2
Use the velocity potential from part (a) and this conformal mapping to
find the velocity potential for the flow of an ideal fluid in the region shown
in Figure 7.89.
y
y
α
Figure 7.88 Figure for Problem 28
Computer Lab Assignments
1
x
–2
Figure 7.89 Figure for Problem 28
In Problems 29–36, use a CAS to plot the isotherms for the given steady-state
temperature φ(x, y).
29. φ(x, y) is the steady-state temperature from Problem 1.
30. φ(x, y) is the steady-state temperature from Problem 2.
31. φ(x, y) is the steady-state temperature from Problem 3.
32. φ(x, y) is the steady-state temperature from Problem 4.
33. φ(x, y) is the steady-state temperature from Problem 5.
34. φ(x, y) is the steady-state temperature from Problem 6.
35. φ(x, y) is the steady-state temperature from Problem 11.
36. φ(x, y) is the steady-state temperature from Problem 12.
2
x