Complex Analysis - Maths KU

Lines of force
ψ = c2 Lower
potential
Equipotential curves
φ = c1 Figure 3.6 Electric field
Equipotential curves
φ = c1 Figure 3.7 Fluid flow
Higher
temperature
y
y
Isotherms
φ = c1 Figure 3.8 Flow of heat
Higher
potential
Streamlines ψ = c2 Flow lines or
flux lines
ψ = c2 Lower
temperature
x
x
3.4 Applications 167
Complex Potential In general, if a potential function φ(x, y) satisfies
Laplace’s equation in some domain D, it is harmonic, and we know from
Section 3.3 that there exists a harmonic conjugate function ψ(x, y) defined
in D so that the complex function
Ω(z) =φ(x, y)+iψ(x, y) (8)
is analytic in D. The function Ω(z) in (8) is called the complex potential
corresponding to the real potential φ. As we have seen in the initial discussion
of this section, the level curves of φ and ψ are orthogonal families. The level
curves of φ, φ(x, y) =c1, are called equipotential curves—that is, curves
along which the potential is constant. In the case in which φ represents
electrostatic potential, the electric field intensity F must be along the family
of curves orthogonal to the equipotential curves because the force of the field
is the gradient of the potential φ, F(x, y) =−∇φ, and as demonstrated in
(6), the gradient vector at a point (x0, y0) is perpendicular to a level curve of
φ at (x0, y0). For this reason the level curves ψ(x, y) =c2, curves that are
orthogonal to the family φ(x, y) =c1, are called lines of force and are the
paths along which a charged particle will move in the electrostatic field. See
Figure 3.6.
Ideal Fluid In fluid mechanics a flow is said to be two-dimensional
or a planar flow if the fluid (which could be water, or even air, moving at
slow speeds) moves in planes parallel to the xy-plane and the motion and
physical properties of the fluid in each plane is precisely the same manner as
it is in the xy-plane. Suppose F(x, y) is the two-dimensional velocity field of
a nonviscous fluid that is incompressible, that is, a fluid for which div F =
0or∇·F =0. The flow is irrotational if curl F = 0 or ∇×F = 0. § An
incompressible fluid whose planar flow is irrotational is said to be an ideal
fluid. The velocity field F of an ideal fluid is a gradient field and can be represented
by (7), where φ is a real-valued function called a velocity potential.
The level curves φ(x, y) =c1 are called equipotential curves or simply
equipotentials. Moreover, φ satisfies Laplace’s equation because div F =0
is equivalent to ∇·F = ∇·(∇φ) =0or∇ 2 φ = 0 and so φ is harmonic. The
harmonic conjugate ψ(x, y) is called the stream function and its level curves
ψ(x, y) =c2 are called streamlines. Streamlines represent the actual paths
along which particles in the fluid will move. The function Ω(z) =φ(x, y)+
iψ(x, y) is called the complex velocity potential of the flow. See Figure
3.7.
Heat Flow Finally, if φ(x, y) represents time-independent or steadystate
temperature that satisfies Laplace’s equation, then the level curves
φ(x, y) = c1 are curves along which the temperature is constant and are
called isotherms. The level curves ψ(x, y) =c2 of the harmonic conjugate
function of φ are the curves along which heat flows and are called flow lines
or flux lines. See Figure 3.8.
§ We will discuss fluid flow in greater detail in Section 5.6.