Many transient transport problems are governed by the same one-dimensional diffusion equation

$$\boxed{\frac{\partial \phi}{\partial t}=D\frac{\partial^2\phi}{\partial x^2}},$$

where $D>0$ is a constant diffusivity and $\phi(x,t)$ is the quantity being transported.

A canonical problem is a semi-infinite domain $x\ge0$ that is initially uniform,

$$\phi(x,0)=\phi_i\qquad(x>0),$$

while its boundary is suddenly changed and then held at

$$\phi(0,t)=\phi_s\qquad(t>0).$$

Far from the boundary, the disturbance has not yet arrived:

$$\phi(x\to\infty,t)\to\phi_i.$$

The same mathematics can describe temperature, concentration, or momentum when the corresponding physical law reduces to diffusion.

Normalize the transported quantity

Define

$$\theta(x,t)=\frac{\phi(x,t)-\phi_i}{\phi_s-\phi_i}.$$

Then

$$\frac{\partial\theta}{\partial t}=D\frac{\partial^2\theta}{\partial x^2},$$

with

$$\theta(0,t)=1,$$

$$\theta(x\to\infty,t)=0,$$

and

$$\theta(x,0)=0\qquad(x>0).$$

The solution depends on distance and time, but the diffusion equation suggests that they enter through the combination

$$x\sim\sqrt{Dt}.$$

Introduce the dimensionless similarity variable

$$\boxed{\eta=\frac{x}{2\sqrt{Dt}}}.$$

Seek a solution of the form

$$\theta(x,t)=f(\eta).$$

Reduce the PDE to an ODE

The similarity variable satisfies

$$\frac{\partial\eta}{\partial x}=\frac{1}{2\sqrt{Dt}},$$

and

$$\frac{\partial\eta}{\partial t}=-\frac{\eta}{2t}.$$

Therefore

$$\frac{\partial\theta}{\partial t} =-\frac{\eta}{2t}f'(\eta),$$

while

$$\frac{\partial^2\theta}{\partial x^2} =\frac{1}{4Dt}f''(\eta).$$

Substitution into the diffusion equation gives

$$-\frac{\eta}{2t}f' =\frac{1}{4t}f'',$$

so

$$\boxed{f''+2\eta f'=0}.$$

The original PDE has collapsed to an ordinary differential equation.

Let

$$g=f'.$$

Then

$$g'+2\eta g=0,$$

whose solution is

$$g=C e^{-\eta^2}.$$

Integrating once more and applying

$$f(0)=1,\qquad f(\infty)=0$$

gives

$$\boxed{f(\eta)=\operatorname{erfc}(\eta)},$$

where the complementary error function is defined by

$$\boxed{\operatorname{erfc}(\eta) =\frac{2}{\sqrt\pi}\int_\eta^\infty e^{-s^2},ds}.$$

Thus the semi-infinite diffusion solution is

$$\boxed{ \frac{\phi(x,t)-\phi_i}{\phi_s-\phi_i} =\operatorname{erfc}!\left(\frac{x}{2\sqrt{Dt}}\right) }.$$

Equivalently,

$$\boxed{ \phi(x,t)=\phi_i+(\phi_s-\phi_i) \operatorname{erfc}!\left(\frac{x}{2\sqrt{Dt}}\right) }.$$

Why the profile is self-similar

At different times the profile spreads, but its shape does not change when distance is measured using

$$\eta=\frac{x}{2\sqrt{Dt}}.$$

Two points at different times with the same value of $x/\sqrt{Dt}$ have the same normalized value $\theta$.

The characteristic penetration distance therefore scales as

$$\boxed{\delta\sim\sqrt{Dt}}.$$

Reaching a fixed multiple of the diffusion depth requires

$$\boxed{t\sim\frac{\delta^2}{D}}.$$

Diffusive penetration grows only with the square root of time: reaching twice as far takes about four times as long.

Worked example

Suppose

$$D=4.0\times10^{-6},\mathrm{m^2/s}$$

and

$$t=25,\mathrm s.$$

Then

$$\sqrt{Dt} =\sqrt{(4.0\times10^{-6})(25)} =1.0\times10^{-2},\mathrm m.$$

At

$$x=2.0,\mathrm{cm}=2\sqrt{Dt},$$

the similarity variable is

$$\eta=\frac{x}{2\sqrt{Dt}}=1.$$

Since

$$\operatorname{erfc}(1)\approx0.157,$$

the local change from the initial value is about $15.7%$ of the imposed boundary change:

$$\frac{\phi-\phi_i}{\phi_s-\phi_i}\approx0.157.$$

The disturbance has reached that location, but it is much weaker there than at the boundary.

What is reusable about this solution

The error-function profile is not specific to one material or one kind of transport. Whenever a physical problem reduces to

$$\phi_t=D\phi_{xx}$$

with a sudden fixed boundary value in a semi-infinite domain, the same similarity solution applies after the physical variables are mapped onto $\phi$ and $D$.

The physics determines what diffuses and the value of the diffusivity; the similarity solution supplies the shared transient spatial structure.