Complex Analysis - Maths KU

5.6 Applications 295
In Problems 5–8, give the complex representation f(z) of the velocity field F(x, y).
Express the function g(z) =f(z) in terms of the symbol z and verify that g(z) isan
analytic function in an appropriate domain D.
5. F(x, y) in Problem 1 6. F(x, y) in Problem 2
7. F(x, y) in Problem 3 8. F(x, y) in Problem 4
In Problems 9–12, find the velocity field F(x, y) of the flow of an ideal fluid determined
by the given analytic function g(z).
9. g(z) =(1+i)z 2
10. g(z) = sin z
11. g(z) =e x cos y + ie x sin y 12. g(z) =x 3 − 3xy 2 + i � 3x 2 y − y 3�
In Problems 13–16, find a complex velocity potential Ω(z) of the complex representation
f(z) of the indicated velocity field F(x, y). Verify your answer using (17).
Describe the equipotential lines and the streamlines.
13. F(x, y) in Problem 1 14. F(x, y) in Problem 2
15. F(x, y) in Problem 3 16. F(x, y) in Problem 4
In Problems 17 and 18, the given analytic function Ω(z) is a complex velocity
potential for the flow of an ideal fluid. Find the velocity field F(x, y) of the flow.
17. Ω(z) = 1
3 iz3
19. Show that
18. Ω(z) = 1
4 z4 + z
��
F(x, y) =A 1 − x2 − y 2
(x2 + y2 ) 2
�
2xy
i −
(x2 + y2 �
j , A > 0,
) 2
is a velocity field for an ideal fluid in any domain D not containing the origin.
�
20. Verify that the analytic function Ω(z) = A z + 1
�
is a complex velocity
z
potential for the flow whose velocity field F(x, y) is in Problem 19.
21. (a) Consider the velocity field in Problem 19. Describe the field F(x, y) ata
point (x, y) far from the origin.
(b) For the complex velocity potential in Problem 20, how does the observation
that Ω(z) → Az as |z| increases verify your answer to part (a)?
22. A stagnation point in a fluid flow is a point at which the velocity field
F(x, y) =0. Find the stagnation points for:
(a) the flow in Example 3(a).
(b) the flow in Problem 19.
23. For any two real numbers k and x1, the function Ω(z) =kLn(z −x1) is analytic
in the upper half-plane and therefore is complex potential for the flow of an
ideal fluid. The real number x1 is a sink when k0.
(a) Show that the streamlines are rays emanating from x1.
(b) Show that the complex representation f(z) of the velocity field F(x, y) of
the flow is
z − x1
f(z) =k
|z − x1| 2
and conclude that the flow is directed toward x1 precisely when k