A Hot Spot Spreading Around a Brake Disc

Brake pads heat one patch of a bike's disc rotor. Once the bike stops, that heat spreads around the ring in both directions. A ring has no ends, so the boundary condition is periodic, and the ring settles to its average temperature.

How it works

Periodic boundary conditions on a ring

In one sentence

On a ring, $u(\theta + 2\pi, t) = u(\theta, t)$ and $u_\theta(\theta + 2\pi, t) = u_\theta(\theta, t)$ . Separation of variables gives the full Fourier series $u = A_0 + \sum e^{-kn^2t/R^2}(A_n\cos n\theta + B_n\sin n\theta)$ with coefficients from the initial temperature, and $u \to A_0$ , the initial average, as $t \to \infty$ .

General formula

u({\htmlData{sym=5}{\theta}}, t) = {\htmlData{sym=0}{A_0}} + \sum_{{\htmlData{sym=6}{n}}=1}^{\infty} e^{-{\htmlData{sym=1}{k}}{\htmlData{sym=6}{n}}^2t/{\htmlData{sym=2}{R}}^2}\left({\htmlData{sym=3}{A_n}}\cos {\htmlData{sym=6}{n}}{\htmlData{sym=5}{\theta}} + {\htmlData{sym=4}{B_n}}\sin {\htmlData{sym=6}{n}}{\htmlData{sym=5}{\theta}}\right)

Heat can't escape a ring with insulated faces; it only evens out. Sharp details fade first, and the average stays forever.

What the symbols mean

$A_0$
Average temperature
The ring's mean temperature: the n = 0 term, which never decays.
$k$
Thermal diffusivity
How fast heat spreads in steel.
$R$
Ring radius
About 8 cm for a bike rotor.
$A_n$
Cosine amplitudes
How much of each cosine pattern the initial temperature contains.
$B_n$
Sine amplitudes
How much of each sine pattern it contains (zero for a patch centered at angle 0).
$\theta$
Angle
Position around the ring.
$n$
Mode number
How many waves fit around the ring.

Unroll the rim into a strip of length $2\pi R$ with its ends glued together. Heat obeys $u_t = \frac{k}{R^2}u_{\theta\theta}$ along it, and the glued ends must match: $u$ and $u_\theta$ are periodic in $\theta$ .
Periodic conditions allow cosines and sines of every whole number of waves around the ring, mode $n$ decaying like $e^{-kn^2t/R^2}$ .
The $n = 0$ term, the average $A_0$ , never decays: with no ends, no heat leaves, so the ring settles at its average temperature.
A rod with insulated ends behaves the same way, with cosine modes only, and also settles at its average. (A real rotor also loses heat to the air, which slowly lowers the average.)

Learn the math

Insulated Ends and Circular RingsPDE · Lesson 8, Separation of Variables