Complex Analysis - Maths KU

Note: Normalized vector fields should
not be used for certain applications.
☞
2.7 Applications 135
Keep in mind that in manyapplications the magnitudes of the vectors are
important and, in such cases, a normalized vector field is inappropriate. We
will see examples of this in Chapter 5 when we discuss the circulation and net
flux of a fluid flow. In this text, whenever we use a normalized vector field,
we will explicitlystate so. Therefore, in a graphical sense, the term vector
field will refer to a plot of a set of vectors that has not been normalized.
Fluid Flow One of the manyuses of vector fields in applied mathematics
is to model fluid flow. Because we are confined to two dimensions in
complex analysis, let us consider only planar flows of a fluid. This means
that the movement of the fluid takes place in planes that are parallel to the
xy-plane and that the motion and the physical traits of the fluid are identical
in all planes. These assumptions allow us to analyze the flow of a single sheet
of the fluid. Suppose that f(z) =P (x, y)+iQ(x, y) represents a velocity
field of a planar flow in the complex plane. Then f(z) specifies the velocity
of a particle of the fluid located at the point z in the plane. The modulus
|f(z)| is the speed of the particle and the vector f(z) gives the direction of
the flow at that point.
For a velocityfield f(z) = P (x, y) +iQ(x, y) of a planar flow, the
functions P and Q represent the components of the velocityin the x- and
y-directions, respectively. If z(t) =x(t) +iy(t) is a parametrization of the
path that a particle follows in the fluid flow, then the tangent vector z ′ (t) =
x ′ (t)+iy ′ (t) to the path must coincide with f(z(t)). Therefore, the real and
imaginaryparts of the tangent vector to the path of a particle in the fluid
must satisfythe system of differential equations
dx
= P (x, y)
dt
dy
= Q (x, y).
dt
The familyof solutions to the system of first-order differential equations (3)
is called the streamlines of the planar flow associated with f(z).
EXAMPLE 2 Streamlines
Find the streamlines of the planar flow associated with f(z) =¯z.
Solution Since f(z) =¯z = x−iy, we identify P (x, y) =x and Q(x, y) =−y.
From (3) the streamlines of f are the familyof solutions to the system of
differential equations
dx
dt
dy
dt
= x
= −y.
(3)