(i) A solute occupying a domain $V_{0}$ has concentration $C(\boldsymbol{x}, t)$ and is created at a rate $S(\boldsymbol{x}, t)$ per unit volume; $\boldsymbol{J}(\boldsymbol{x}, t)$ is the flux of solute per unit area; $\boldsymbol{x}, t$ are position and time. Derive the transport equation

$$C_{t}+\nabla \cdot \boldsymbol{J}=S$$

State Fick's Law of diffusion and hence write down the diffusion equation for $C(\boldsymbol{x}, t)$ for a case in which the solute flux occurs solely by diffusion, with diffusivity $D(\boldsymbol{x})$.

In a finite domain $0 \leqslant x \leqslant L, D, S$ and the steady-state distribution of $C$ depend only on $x ; C$ is equal to $C_{0}$ at $x=0$ and $C_{1} \neq C_{0}$ at $x=L$. Find $C(x)$ in the following two cases: (a) $D=D_{0}, S=0$, (b) $D=D_{1} x^{1 / 2}, S=0$,

where $D_{0}$ and $D_{1}$ are positive constants.

Show that there is no steady solution satisfying the boundary conditions if $D=$ $D_{1} x, S=0 .$

(ii) For the problem of Part (i), consider the case $D=D_{0}, S=k C$, where $D_{0}$ and $k$ are positive constants. Calculate the steady-state solution, $C=C_{s}(x)$, assuming that $\sqrt{k / D_{0}} \neq n \pi / L$ for any integer $n$.

Now let

$$C(x, 0)=C_{0} \frac{\sin \alpha(L-x)}{\sin \alpha L}$$

where $\alpha=\sqrt{k / D_{0}}$. Find the equations, boundary and initial conditions satisfied by $C^{\prime}(x, t)=C(x, t)-C_{s}(x)$. Solve the problem using separation of variables and show that

$$C^{\prime}(x, t)=\sum_{n=1}^{\infty} A_{n} \sin \frac{n \pi x}{L} \exp \left[\left(\alpha^{2}-\frac{n^{2} \pi^{2}}{L^{2}}\right) D_{0} t\right]$$

for some constants $A_{n}$. Write down an integral expression for $A_{n}$, show that

$$A_{1}=-\frac{2 \pi C_{1}}{\alpha^{2} L^{2}-\pi^{2}},$$

and comment on the behaviour of the solution for large times in the two cases $\alpha L<\pi$ and $\alpha L>\pi$.