Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: GENERAL AND PARTICULAR SOLUTIONS

20.1.2 The diffusion equation

The diffusion equation

κ∇^2 u=

∂u

∂t

(20.2)

describes the temperatureuin a region containing no heat sources or sinks; it

also applies to the diffusion of a chemical that has a concentrationu(r,t). The

constantκis called the diffusivity. The equation is clearly second order in the

three spatial variables, but first order in time.

Derive the equation satisfied by the temperatureu(r,t)at timetfor a material of uniform

thermal conductivityk, specific heat capacitysand densityρ. Express the equation in

Cartesian coordinates.

Let us consider an arbitrary volumeVlying within the solid and bounded by a surfaceS
(this may coincide with the surface of the solid if so desired). At any point in the solid
the rate of heat flow per unit area in any given directionrˆis proportional to minus the
component of the temperature gradient in that direction and so is given by (−k∇u)·ˆr.The
total flux of heatoutof the volumeVper unit time is given by

−

dQ

dt

=

∫∫

S

(−k∇u)·nˆdS

=

∫∫∫

V

∇·(−k∇u)dV , (20.3)

whereQis the total heat energy inVat timetandnˆis the outward-pointing unit normal
toS; note that we have used the divergence theorem to convert the surface integral into
a volume integral.
We can also expressQas a volume integral overV,

Q=

∫∫∫

V

sρu dV ,

and its rate of change is then given by

dQ

dt

=

∫∫∫

V

sρ

∂u

∂t

dV , (20.4)

where we have taken the derivative with respect to time inside the integral (see section 5.12).
Comparing (20.3) and (20.4), and remembering that the volumeVis arbitrary, we obtain
the three-dimensional diffusion equation

κ∇^2 u=

∂u

∂t

,

where the diffusion coefficientκ=k/(sρ). To express this equation in Cartesian coordinates,
we simply write∇^2 in terms ofx,yandzto obtain

κ

(

∂^2 u

∂x^2

+

∂^2 u

∂y^2

+

∂^2 u

∂z^2

)

=

∂u

∂t

.

The diffusion equation just derived can be generalised to

k∇^2 u+f(r,t)=sρ

∂u

∂t

.