284 Chapter 4 The Potential Equation
Figure 3 Streamlines (solid) and equipotential curves (dashed) for flow in a cor-
ner. The streamlines are described by the equation 2xy=constant, with a differ-
ent constant for each one. Similarly, the equipotential curves are described by the
equationx^2 −y^2 =constant.
It turns out that the potential equation (but not all elliptic equations) is
best studied through the use of complex variables. A complex variable may be
writtenz=x+iy,wherexandyare real andi^2 =−1; similarly a function of
zis denoted byf(z)=u(x,y)+iv(x,y),uandvbeing real functions of real
variables. Iffhas a derivative with respect toz,thenbothuandvsatisfy the
potential equation. Easy examples, such as polynomials and exponentials, lead
to familiar solutions:
z^2 =(x+iy)^2 =x^2 −y^2 +i 2 xy,
ez=ex+iy=exeiy=excos(y)+iexsin(y),ln(z)=^1
2ln(
x^2 +y^2)
+itan−^1(y
x)
.
(See Section 4.1, Exercises 1, 2; Section 4.4, Exercises 13, 14; Miscellaneous
Exercise 18 in this chapter.) Knowing these elementary solutions often helps
in simplifying a problem.
In certain idealized fluid flows (steady, irrotational, two-dimensional flow of
an inviscid, incompressible fluid) the velocity vector is given byV=−gradφ,
where thevelocity potentialφis a solution of the potential equation. The
streamlines along which the fluid flows are level curves of a related functionψ,
called thestream function, which also is a solution of the potential equation.
The two functionsφandψare, respectively, the real and imaginary parts
of a function of the complex variablez. The level curvesφ=constant and
ψ=constant form two families of orthogonal curves called aflow net.The
flow net in Fig. 3, forφ=x^2 −y^2 andψ= 2 xy(the real and imaginary parts
of the functionf(z)=z^2 ), illustrates flow near a corner formed by two walls.
Many other flow nets are shown in the bookPotential Flows:Computer Graphic
Solutionsby R.H. Kirchhoff. Civil engineers sometimes sketch a flow net by eye