18.3
Heat Problems
Laplace’s equation also governs steady heat problems (cf. Sections 12.5, 12.6):
the heat equation is given by
T
t
=
c
2
∆
T
,where
T
[K] denotes the tempera
ture and where
c
[m
2
s

1
] denotes the thermal diFusivity. We have
T
t
≡
0in
steady state, i. e. Laplace’s equation. Therefore in two space dimensions, we
may again use methods from complex analysis. In this application,
T
is also
called the
heat potential,
and it is the real part of the
complex heat potential
F
(
z
)=
T
(
x, y
)+
i
Ψ(
x, y
)
.
(1040)
The curves
T
≡
const
.
, are called
isotherms
and the curves Ψ
≡
const
.
are
called
heat fow lines.
Heat ±ows along these lines from higher to lower tem
peratures.
Examples:
The examples considered in the previous section can now be rein
terpreted as problems on heat ±ow: the electrostatic potential lines now
become isotherms, and the lines of electrical force become lines of heat ±ow.
So we can immediately write down the heat potential between two plates, two
cylinders and so on. We consider two new examples with diFerent boundary
conditions:
1. We consider the ²rst quadrant of the unit disk. The temperature on
the straight line segments is prescribed (Dirichlet boundary conditions),
but on the arc, the domain is assumed to be insulated, so that
∂
n
T
=0
(Neumann boundary condition). We use polar coordinates (
r, ϑ
), so
that
∂
n
≡
∂
r
on the arc. The angular solution from Example 3 before
satis²es the Neumann boundary condition on the arc. The complex
potential is thus given by
F
(
z
a

ib
Log
z
with real part
T
(
x, y
a
+
bϑ
,
ϑ
=Im(Log
z
). Coeﬃcients
a, b
are determined from the Dirichlet
boundary values.
2. We consider the upper halfplane. On the boundary (real axis), the
temperature is prescribed for
x<

1and
x>
1(twod
iFerentva
lues)
and the domain is insulated on (

1
,
1). The upper half of the
z
plane
is mapped onto a semiin²nite vertical strip in the
w
plane,
S
:=
{
w
∈
C

u
∈
(

π/
2
,π/
2)
,v
≥
0
}
, by the inverse of the sine mapping:
w
=
f
(
z
)=arcs
in
z.