Dimensional analysis can also be used to solve certain types of partial differential equations. If this seems too good to be true, it isn't. Here we will concentrate on the solution of the diffusion equation; we will encounter this equation many times in the remainder of the course, so it will be useful to work out some of its properties now.

We'll start by deriving the one-dimensional diffusion, or heat, equation.^{1} Let represent the temperature of a metal bar
at a point *x* at time *t* (I'll use to avoid confusion with the
symbol for the dimension of time, *T*). The first step is the derivation
of a *continuity equation*
for the heat flow
in the bar. Let the bar have a cross sectional area *A*, so that the
infinitesimal volume of the bar between *x* and is . The quantity of heat contained in this volume is ,
with the specific heat at constant pressure *per unit volume*; it
has dimensions . In a time interval *dt*
this heat changes by an amount due to the change in temperature. This change in the heat must come from
somewhere, and is the result of a flux of heat *q*(*x*,*t*) through the area *A*
(*q* is the heat flowing through a unit area per unit time). Into the left
side of the volume an amount of heat *q A dt* flows in a time *dt*; on the
right hand side of the volume a quantity flows out in a time *dt*, so that the net accumulation of heat
in the volume is . Equating the two
expressions for the rate of change of the heat in the volume ,
we find

which is the equation of continuity. It is a mathematical expression of the
conservation of heat in the infinitesimal volume . We
supplement this with a phenomenological law of heat conduction, known as
*Fourier's law*:
the heat flux is proportional
to the negative of the local temperature gradient (heat flows from a hot
reservoir to a cold reservoir):

with the *thermal conductivity*
of the metal bar. The thermal conductivity is usually measured in units of , and has dimensions . See the Table on p. 118 of the PQRG for the thermal
conductivities of some materials. Combining Eqs. (1.22) and (1.23), we obtain the
*diffusion equation*
(often called the *heat equation*)

where is the *thermal diffusivity* of the metal bar; it
has dimensions , as it should. Eq. (1.24) is the diffusion equation for heat. The diffusion equation will
appear in many other contexts during this course. It usually results from
combining a continuity equation with an empirical law which expresses a
current or flux in terms of some local gradient.

Suppose that the bar is very long, so that we can consider the idealized
case of an infinite bar. At an initial time *t*=0, we add an amount of heat *H* (with dimensions ) at some point of the bar, which we
will arbitrarily call *x*=0. We could do this, for instance, by briefly
holding a match to the bar. The heat is conserved at all times, so that

How does this heat diffuse away from *x*=0 as a function of time *t*; i.e.,
what is ? We first identify the important parameters. The
temperature certainly depends upon *x*, *t*, and the diffusivity *D*; we see from Eq. (1.25) that it also depends upon the initial
conditions through the combination . What are the
dimensions? We have [*x*]=*L*, [*t*]=*T*, , and ,
so that *n*=4. These dimensions are not independent, for the quantity is dimensionless, so that *k*=3. We will choose as our
independent quantities (*t*,*D*,*Q*). Now express in terms of these
variables:

We find

which has the solution *a*=-1/2, *b*=-1/2, *c*=1. Therefore, dimensional
analysis tells us that the solution of the diffusion equation is of the form

with a function which we still need to determine. The important
point is that is only a function of the combination ,
and not *x* and *t* separately. To determine , let's introduce the
dimensionless variable . Now use the chain rule to calculate
various derivatives of :

Substituting Eqs. (1.30) and (1.31) into the diffusion equation (1.24), and canceling various factors, we obtain a differential equation for ,

*Dimensional analysis has reduced the problem from the solution of a
partial differential equation in two variables to the solution of an
ordinary differential equation in one variable!* The normalization
condition, Eq. (1.25), becomes in these variables

You might think that Eq. (1.32) is hard to solve; however, it turns out that it is an exact differential,

which we can integrate once to obtain

However, since any physically reasonable solution would have both and as , the
integration constant must be zero. We now need to solve a first order
differential equation, which we do by dividing Eq. (1.35) by , multiplying by *dz*, and integrating, with the result that , or

with *C* a constant. To determine *C*, we use the normalization condition,
Eq, (1.33):

where the integral (known as a *Gaussian integral*) can be found in
integral tables. Therefore . Returning to our original variables, we have

This is the complete solution for the temperature distribution in a
one-dimensional bar due to a point source of heat
^{2}. Here are pictures:

**Click here** to see an animation (controlled
by mouse buttons) of the one-dimensional diffusion process described
by eq. (1.38). The animation starts at some arbitrary
time *t _{0}* (zero time corresponds to an infinitely sharp
peak) and ends at 25

Footnotes:

^{1}
I prefer the term diffusion equation, since we are just describing the diffusion of heat.

^{2}
For the mathematically sophisticated, I'll mention that the same solution can be obtained using the method of Fourier transforms applied to the diffusion equation.

Thu Jul 10 16:27:59 EDT 1997