(i) The diffusion equation for a spherically-symmetric concentration field $C(r, t)$ is

$$C_{t}=\frac{D}{r^{2}}\left(r^{2} C_{r}\right)_{r},$$

where $r$ is the radial coordinate. Find and sketch the similarity solution to (1) which satisfies $C \rightarrow 0$ as $r \rightarrow \infty$ and $\int_{0}^{\infty} 4 \pi r^{2} C(r, t) d r=M=$ constant, assuming it to be of the form

$$C=\frac{M}{(D t)^{a}} F(\eta), \quad \eta=\frac{r}{(D t)^{b}},$$

where $a$ and $b$ are numbers to be found.

(ii) A two-dimensional piece of heat-conducting material occupies the region $a \leqslant r \leqslant$ $b,-\pi / 2 \leqslant \theta \leqslant \pi / 2$ (in plane polar coordinates). The surfaces $r=a, \theta=-\pi / 2, \theta=\pi / 2$ are maintained at a constant temperature $T_{1}$; at the surface $r=b$ the boundary condition on temperature $T(r, \theta)$ is

$$T_{r}+\beta T=0,$$

where $\beta>0$ is a constant. Show that the temperature, which satisfies the steady heat conduction equation

$$T_{r r}+\frac{1}{r} T_{r}+\frac{1}{r^{2}} T_{\theta \theta}=0,$$

is given by a Fourier series of the form

$$\frac{T}{T_{1}}=K+\sum_{n=0}^{\infty} \cos \left(\alpha_{n} \theta\right)\left[A_{n}\left(\frac{r}{a}\right)^{2 n+1}+B_{n}\left(\frac{a}{r}\right)^{2 n+1}\right]$$

where $K, \alpha_{n}, A_{n}, B_{n}$ are to be found.

In the limits $a / b \ll 1$ and $\beta b \ll 1$, show that

$$\int_{-\pi / 2}^{\pi / 2} T_{r} r d \theta \approx-\pi \beta b T_{1}$$

given that

$$\sum_{n=0}^{\infty} \frac{1}{(2 n+1)^{2}}=\frac{\pi^{2}}{8} .$$

Explain how, in these limits, you could have obtained this result much more simply.