2
Diffusion equations
[Diffusion is the motion of an entity (a substance, heat, a biological species, etc.) from
higher concentrations to lower concentrations. There are many physical processes that
amounts to diffusion of some sort. In this section, we shall consider only heat. However,
the basic ideas for the diffusion of other types of entities are the same, and the same
partial differential equation results from the derivation. ]
2.1
Heat and temperature
Consider a body of mass m where heat Q is conveyed to the body from an external heat source.
Its temperature will rise as more and more heat
is transferred. Figure 2.1 shows a typical example
of how the temperature rises with increasing heat.
There are often plateau regions in the graph where
heat is used to effect a phase change instead of a
temperature rise. A phase change is a change the
structure of the material such as melting, vaporising,
or allotropic transformations (changing of crystal
structure). However, between these plateau regions
the graph normally rises in a fairly linear fashion.
We shall derive the heat equation for materials only
in the temperature range that stays in a linear part of
such a graph.
Figure 2.1
The amount of increase in temperature ∆T in a body of mass m is determined by its
increase in heat ∆Q by means of the following formula
∆Q = cm∆T,
(2)
where c is a constant determined by the material out of which the body is composed. It
is called the specific heat constant. If Q is in joules, T in kelvin, and m in kg, then c is
measured in J.K−1 .kg−1 .
Let us rewrite equation (2) as
Q = mcT + Q0 ,
(3)
where Q0 denotes the amount of heat contained in the body at temperature T = 0.
Note that each linear section of the (Q, T )-graph must be specified by its own constants
c and Q0 . Also the value of Q0 depends on which temperature scale is used (Kelvin,
Fahrenheit, etc.).
2
DIFFUSION EQUATIONS
2.2
8
Heat flow due to a temperature gradient
Consider a uniform rod with cross section area A, and consider a plane section
through the rod at some point x. If there
is a temperature difference from one side
of the plane to the other, heat will flow
across the plane because of the temperature gradient. This may be expressed as
∂Q
∂T
= −KA
,
∂t
∂x
(4)
where K is a constant that describes how
well the material conducts heat. It is
called the heat conductivity. The equation
(4) is a variant of Fick’s first law, that
states that the flux of a substance is proportional to the negative of its concentration gradient.
Figure 2.2
The minus sign in equation (4) is to make sure that heat flows to the right when the
temperature gradient is negative, and that it flows to the left when the gradient is positive.
In other words, heat will always flow from higher temperatures to lower temperatures,
as we would expect.
2.3
Heat diffusion in 1D
We shall denote the temperature at position x in a long thin rod at time t by u(x, t).
The cross sectional area of the rod is A and it is made from a material with specific heat
constant c, and density ρ.
The proportionality between the heat content Q(x, t) of a small section of the rod centered
at a point x at time t and the temperature measured at that point u(x, t) is given by (2)
as
Q = cmu + Q0 .
(5)
Figure 2.3
Consider a small element of the rod with length ∆x, starting at x and extending to
2
DIFFUSION EQUATIONS
9
x + ∆x. The (small) mass of this element is
∆m = ρA∆x.
(6)
The rate of increase in the (small amount of) heat in this element, denoted by (∆Q)t ,
comes from heat flowing in over both sides of the element. Therefore
(∆Q)t = (∆Qleft edge )t + (∆Qright edge )t
Note that (∆Qleft edge )t comprises heat flowing to the right , while (∆Qright edge )t comprises heat flowing to the left. These are therefore given by
(∆Qleft edge )t = −KAux (x, t)
(∆Qright edge )t = +KAux (x + ∆x, t)
Therefore
(∆Q)t = −KAux (x, t) + KAux (x + ∆x, t).
(7)
It seems plausible that the total heat content in the small element may be approximated
using equation (5), where the temperature is taken right in the middle of the element,
∆Q = c(∆m)u(x + 12 ∆x, t) + Q0 .
(8)
We take the time derivative of (8) because the left hand side of equation (7) requires it,
(∆Q)t =
´
∂ ³
cmu(x + 21 ∆x, t)) + Q0 = c∆mu(x + 21 ∆x, t)t .
∂t
Substituting ∆m from (6), gives
(∆Q)t = (cρA∆x)ut (x + 21 ∆x, t).
Substitution of this expression into the left hand side of (7) gives
(cρA∆x)ut (x + 12 ∆x, t) = KA[ux (x + ∆x, t) − ux (x, t)],
or
µ
ut (x + 12 ∆x, t) =
Let κ =
then
K
cρ
¶µ
¶
ux (x + ∆x, t) − ux (x, t)
.
∆x
K
, and take the limit as the length of the element tends to zero, i.e. as ∆x → 0,
cρ
ut = σ uxx .
(9)
Equation (9) is known as the Heat equation in one dimension. It is also called the
Diffusion equation, or Fick’s second law.
2
DIFFUSION EQUATIONS
2.4
10
Heat equation in 2D
Consider a plate of uniform thickness h, specific heat constant c, and density ρ. We
consider a small rectangular element of size ∆x by ∆y, with its bottom left corner lying
at (x, y).
Figure 2
The rate of increase in the heat in this element, denoted by (∆Q)t comes from heat
flowing into the element on all four sides as shown in the figure,
(∆Q)t = (∆Q(1) )t + (∆Q(2) )t + (∆Q(3) )t + (∆Q(4) )t .
(10)
(∆Q(1) )t flows to the right over an area with size h∆y, and we assume that the temperature gradient in the middle of this area, at (x, y + 21 ∆y), determines this flow, therefore
(∆Q(1) )t = −K(h∆y)ux (x, y + 12 ∆y, t).
But (∆Q(2) )t flows to the left over the same area, but the temperature gradient at
(x + ∆x, y + 12 ∆y) determines this flow, therefore
(∆Q(2 )t = +K(h∆y)ux (x + ∆x, y + 21 ∆y, t).
Similarly (∆Q(3) )t flows upwards, and depends uy (x + 12 ∆x, y, t), while (∆Q(4) )t flows
downwards, and depends on uy (x + 12 ∆x, y + ∆y, t), therefore
(∆Q(3) )t = −K(h∆x)uy (x, y + 21 ∆y, t),
and
(∆Q(4 )t = +K(h∆x)uy (x + ∆x, y + 21 ∆y, t).
Then
h
(∆Q)t = Kh −∆yux (x, y + 21 ∆y, t) + ∆yux (x + ∆x, y + 21 ∆y, t)
i
−∆xuy (x, y + 12 ∆y, t) + ∆xuy (x + ∆x, y + 21 ∆y, t) .
(11)
But, from (5),
∆Q = cu∆m = (cρh∆x∆y)u,
(12)
2
DIFFUSION EQUATIONS
11
and taking the time derivative of (12), gives
(∆Q)t = (cρh∆x∆y)ut .
(13)
Equating (11) and (13), gives
h
cρh∆x∆yut = Kh −∆yux (x, y + 12 ∆y, t) + ∆yux (x + ∆x, y + 21 ∆y, t)
i
−∆xuy (x, y + 21 ∆y, t) + ∆xuy (x + ∆x, y + 21 ∆y, t) .
After dividing by cρh∆x∆y, we obtain
"
ut
#
K ux (x + ∆x, y + 21 ∆y, t) − ux (x, y + 12 ∆y, t)
=
cρ
∆x
"
#
1
uy (x + ∆x, y + 2 ∆y, t) − uy (x, y + 21 ∆y, t)
+
.
∆y
(14)
Let the size of the element shrink, i.e. take the limit as ∆x → 0, and ∆y → 0, then (14)
becomes
ut = κ(uxx + uyy ),
(15)
where κ =
K
.
cρ
Equation (15) is the two dimensional heat (diffusion) equation. It models the temperature
evolution in a uniform plate.
In the limit as t → ∞, the temperature profile in the plate should stabilize, so that
eventually u(x, y, t) no longer changes with time, i.e. ut = 0. This condition is referred
to as the steady state, and then (15) simplifies to
uxx + uyy = 0,
which is Laplace’s equation in x and y.
(16)