Complex Analysis - Maths KU

y
Figure 7.56 Flow around a corner
Figure 7.57 Flow around a cylinder
y
i
x
x
7.5 Applications 439
flow of an ideal fluid in the first quadrant. Since Ω(z) =z 2 = x 2 − y 2 +2xyi,
the streamlines of this flow are the curves 2xy = c2. Some streamlines have
been plotted in Figure 7.56. It should be clear from this figure whythis flow
is referred to as “flow around a corner.”
EXAMPLE 5 Flow arounda Cylinder
Construct a flow of an ideal fluid in the domain consisting of all points outside
the unit circle |z| = 1 and in the upper half-plane y>0 shown in Figure 7.57.
Solution Let D be the domain shown in Figure 7.57. Identifying a =2in
entryH-3 of Appendix III, we obtain the one-to-one conformal mapping
w = Ω(z) =z + 1
z
of D onto the upper half-plane v>0. In addition, entryH-3 indicates that
the boundaryof D is mapped onto the real axis v = 0. Therefore,
f(z) =Ω ′ (z) =1 − 1 1
=1−
z2 ¯z 2
is a complex representation of a flow of an ideal fluid in D. Since
Ω(z) =z + 1
= x +
z
x
x2 + i
+ y2 the streamlines of this flow are the curves
ψ(x, y) =c2, or y −
�
y −
y
x 2 + y 2
y
x2 = c2.
+ y2 Some streamlines for this flow have been plotted in Figure 7.57.
It is not always possible to describe the streamlines with a Cartesian
equation in the variables x and y. This situation occurs when an appropriate
mapping z = Ω −1 (w) of a domain D ′ in the w -plane onto the domain D in
the z-plane can be found, but you cannot solve for the mapping w = Ω(z).
In such cases, it is possible to describe the streamlines parametrically.
EXAMPLE 6 Streamlines DefinedParametrically
Construct a flow of an ideal fluid in the domain D consisting of all points in the
upper half-plane y>0 excluding the points on the ray y = π, −∞ 0 onto the domain D.
�
,