Complex Analysis - Maths KU

296 Chapter 5 Integration in the Complex Plane
–1 1
Figure 5.57 Figure for Problem 24
x
24. The complex potential Ω(z) =kLn(z − 1) − kLn(z +1), k>0, determines
the flow of an ideal fluid on the upper half-plane y>0 with a single source at
z = 1 and a single sink at z = −1. Show that the streamlines are the family of
circles x 2 +(y − c2) 2 =1+c 2 2. See Figure 5.57.
In Problems 25–30, compute the circulation and net flux for the given flow and the
indicated closed contour C.
25. f(z) = 1
; where C is the circle |z| =1
z
26. f(z) =2z; where C is the circle |z| =1
27. f(z) = 1
; where C is the circle |z − 1| =2
z − 1
28. f(z) =¯z; where C is the square with vertices z =0, z =1, z =1+i, z = i
29. F(x, y) =(4x +3y)i +(2x − y)j, where C is the circle x 2 + y 2 =4
30. F(x, y) =(x +2y)i +(x − y)j, where C is the square with vertices z =0,
z =1+i, z =2i, z = −1+i
Focus on Concepts
31. Suppose f(z) =P (x, y)+iQ(x, y) is a complex representation of a velocity field
F of the flow of an ideal fluid on a simply connected domain D of the complex
plane. Assume P and Q have continuous partial derivatives throughout D. IfC
is any simple closed curve C lying within D, show that the circulation around
C and the net flux across C are zero.
32. The flow described by the velocity field f(z) =(a + ib)/z is said to have a
vortex at z = 0. The geometric nature of the streamlines depends on the
choice of a and b.
(a) Show that if z(t) =x(t)+iy(t) is the path of a particle in the flow, then
dx
dt
dy
dt
= ax − by
x 2 + y 2
bx + ay
=
x2 .
+ y2 (b) Rectangular and polar coordinates are related by r 2 = x 2 +y 2 , tan θ = y/x.
Use these equations to show that
�
dr 1
= x
dt r
dx
�
dy
+ y ,
dt dt
dθ 1
=
dt r2 �
−y dx
�
dy
+ x .
dt dt
(c) Use the equations in parts (a) and (b) to establish that
dr
dt
a dθ
= ,
r dt
b
= .
r2 (d) Use the equations in part (c) to conclude that the streamlines of the flow
are logarithmic spirals r = ce aθ/b , b �= 0. Use a graphing utility to verify
that a particle traverses a path in a counterclockwise direction if and only
if a