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4.6 Heat Flow

We let u(x, y, z, t) be the temperature and let H(t) be the amount of heat (in calories,
say) contained in a region D. Then
ZZZ
H(t) = cρudxdydz,
D

where c is the ”specific heat” of the material and ρ is its density (mass per unit volume).
The change in heat is
ZZZ
dH
= cρu t dxdydz.
dt D
Fourier’s law says that heat flows from hot to cold regions proportionately to the tempera-
ture gradient. But heat cannot be lost from D except by leaving it through the boundary.
This is the law of conservation of energy.Therefore, the change of heat energy in D also
equals the heat flux across the boundary,
ZZ
dH
= k(n · ∇u)dS,
dt G

where k is a proportionality factor (the ”heat conductivity”). By the divergence theorem,
ZZZ ZZZ
∂u
cρ dxdydz = ∇ · (k∇u)dxdydz
D ∂t D
and we get the heat equation
∂u
cρ = ∇ · (k∇u).
∂t
If c, ρ, and k are constants, it is exactly the same as the diffusion equation!

4.7 Stationary Waves and Diffusions

Consider any of the four preceding examples in a situation where the physical state does
not change with time. Then u t = u tt = 0. So both the wave and the diffusion equations
reduce to
∆u = u xx + u yy + u zz = 0.

This is called the Laplace equation. Its solutions are called harmonic functions. For
example, consider a hot object that is constantly heated in an oven. The heat is not
expected to be evenly distributed throughout the oven. The temperature of the object
eventually reaches a steady (or equilibrium) state. This is a harmonic function u(x, y, z).
(Of course, if the heat were being supplied evenly in all directions, the steady state would
be u ≡ constant.) In the one-dimensional case (e.g., a laterally insulated thin rod that
exchanges heat with its environment only through its ends), we would have u a function
of x only. So the Laplace equation would reduce simply to u xx = 0. Hence u = c 1 x + c 2 .
The two- and three-dimensional cases are much more interesting.

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