Introduction - Heat Conduction: Mathematical Models and Analytical Solutions - page 18

Laplace equation or the harmonic equation . For the two-dimensional case,
u , we have the corresponding two-dimensional potential equations.

Continuous solutions of Laplace equations are called harmonic functions .Har-
monic functions of two variables have a close relation with analytical functions in
function of complex variables. Both real and imaginary parts of an analytical func-
tion are harmonic. For any given harmonic function, we may construct an analytical
function. These have very important applications in the theory of functions of com-
plex variables and fluid mechanics.

= u ( x , y )

Electric Potential in an Electrostatic Field

Let E be the electric field intensity, and ρ the charge density. By the Gauss theorem,

we have

i ∂
∂

j ∂
∂

k ∂
∂

∇ · E = 4 πρ , ∇ =

x + y + z .

Since the electrostatic fields are irrotational, there exists an electric potential v such
that E = − ∇ v . Substituting it into ∇ · E = 4 πρ yields Δ v = − 4 πρ .When ρ = 0, we

have 0.

Therefore the electric potential in an electrostatic field is also called a potential
function .

Δ v =

1.3 Theory of Heat Conduction And Three Types
of Heat-Conduction Equations

1.3.1 Constitutive Relations of Heat Flux

By the second law of thermodynamics, there exists a physical quantity Q that is, at
a given time instant, associated with each surface in a non-isothermal body. This
quantity can be interpreted as the heat through the surface and has two fundamental
properties: behaving additively on compatible material surfaces and satisfying the
first law of thermodynamics (the conservation of energy). These two properties,
when rendered precisely, imply the existence of the flux vector field q whose scalar
product with the unit normal vector to the surface yields the surface density of the
heat Q (Šilhavy 1985). q is therefore named the heat-flux density vector ,orthe heat
flux for short.
The relation between the heat flux q and the temperature gradient

T is called
the constitutive relation of heat flux ,orthe constitutive relation for short. It is the
most fundamental and important relation in heat conduction, and is normally given
by fundamental laws.

∇