(a) Solute diffuses and is advected in a moving fluid. Derive the transport equation and deduce that the solute concentration $C(\mathbf{x}, t)$ satisfies the advection-diffusion equation

$$C_{t}+\nabla \cdot(\mathbf{u} C)=\nabla \cdot(D \nabla C)$$

where $\mathbf{u}$ is the velocity field and $D$ the diffusivity. Write down the form this equation takes when $\nabla \cdot \mathbf{u}=0$, both $\mathbf{u}$ and $\nabla C$ are unidirectional, in the $x$-direction, and $D$ is a constant.

(b) A solution occupies the region $x \geqslant 0$, bounded by a semi-permeable membrane at $x=0$ across which fluid passes (by osmosis) with velocity

$$u=-k\left(C_{1}-C(0, t)\right)$$

where $k$ is a positive constant, $C_{1}$ is a fixed uniform solute concentration in the region $x<0$, and $C(x, t)$ is the solute concentration in the fluid. The membrane does not allow solute to pass across $x=0$, and the concentration at $x=L$ is a fixed value $C_{L}$ (where $\left.C_{1}>C_{L}>0\right)$.

Write down the differential equation and boundary conditions to be satisfied by $C$ in a steady state. Make the equations non-dimensional by using the substitutions

$$X=\frac{x k C_{1}}{D}, \quad \theta(X)=\frac{C(x)}{C_{1}}, \quad \theta_{L}=\frac{C_{L}}{C_{1}},$$

and show that the concentration distribution is given by

$$\theta(X)=\theta_{L} \exp \left[\left(1-\theta_{0}\right)(\Lambda-X)\right]$$

where $\Lambda$ and $\theta_{0}$ should be defined, and $\theta_{0}$ is given by the transcendental equation

$$\theta_{0}=\theta_{L} e^{\Lambda-\Lambda \theta_{0}}$$

What is the dimensional fluid velocity $u$, in terms of $\theta_{0} ?$

(c) Show that if, instead of taking a finite value of $L$, you had tried to take $L$ infinite, then you would have been unable to solve for $\theta$ unless $\theta_{L}=0$, but in that case there would be no way of determining $\theta_{0}$.

(d) Find asymptotic expansions for $\theta_{0}$ from equation $(*)$ in the following limits:

(i) For $\theta_{L} \rightarrow 0, \Lambda$ fixed, expand $\theta_{0}$ as a power series in $\theta_{L}$, and equate coefficients to show that

$$\theta_{0} \sim e^{\Lambda} \theta_{L}-\Lambda e^{2 \Lambda} \theta_{L}^{2}+O\left(\theta_{L}^{3}\right) .$$

(ii) For $\Lambda \rightarrow \infty, \theta_{L}$ fixed, take logarithms, expand $\theta_{0}$ as a power series in $1 / \Lambda$, and show that

$$\theta_{0} \sim 1+\frac{\log \theta_{L}}{\Lambda}+O\left(\frac{1}{\Lambda^{2}}\right)$$

What is the limiting value of $\theta_{0}$ in the limits (i) and (ii)?

(e) Both the expansions in (d) break down when $\theta_{L}=O\left(e^{-\Lambda}\right)$. To investigate the double limit $\Lambda \rightarrow \infty, \theta_{L} \rightarrow 0$, show that $(*)$ can be written as

$$\lambda=\phi e^{\phi}$$

where $\phi=\Lambda \theta_{0}$ and $\lambda$ is to be determined. Show that $\phi \sim \lambda-\lambda^{2}+\ldots$ for $\lambda \ll 1$, and $\phi \sim \log \lambda-\log \log \lambda+\ldots$ for $\lambda \gg 1$.

Briefly discuss the implication of your results for the problem raised in (c) above.