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4.5 Diffusion

Let us imagine a motionless liquid filling a straight tube or pipe and a chemical substance,
say a dye, which is diffusing through the liquid. Simple diffusion is characterized by
the following law. [It is not to be confused with convection (transport), which refers to
currents in the liquid.] The dye moves from regions of higher concentration to regions
of lower concentration. The rate of motion is proportional to the concentration gradient.
(This is known as Fick’s law of diffusion.) Let u(x, t) be the concentration (mass per unit
length) of the dye at position x of the pipe at time t.
In the section of pipe from x 0 to x 1 (see Figure 4.5), the mass of dye is

Z Z
x 1 dM x 1
M(t) = u(x, t)dx, so = u t (x, t)dx.
dt
x 0 x 0

x 0 x 1

Figure 4.5: Section of pipe

The mass in this section of pipe cannot change except by flowing in or out of its ends.
By Fick’s law

dM
= flow in − flow out = ku x (x 1 , t) − ku x (x 0 , t), dt
dt
where k is a proportionally constant. Therefore, those two expressions are equal:
Z
x 1
u t (x, t)dx = ku x (x 1 , t) − ku x (x 0 , t).
x 0
Differentiating with respect to x 1 , we get
u t = ku xx .

This is the diffusion equation. In three dimensions we have
ZZZ ZZ
u t dxdydz = k(n · ∇u)dS,
D G
where D is any solid domain and G is its bounding surface. By the divergence theorem
(using the arbitrariness of D as above), we get the three-dimensional diffusion equation
u t = k(u xx + u yy + u zz ) = k∆u.

If there is an external source (or a ”sink”) of the dye, and if the rate k of diffusion is a
variable, we get the more general inhomogeneous equation
u t = ∇ · (k∇u) + f(x, t).

The same equation describes the conduction of heat, brownian/notion, diffusion models
of population dynamics, and many other phenomena.

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