(i) When a solid crystal grows into a supercooled infinite melt, latent heat must be removed from the interface by diffusion into the melt. Write down the equation and boundary conditions satisfied by the temperature $\theta(\boldsymbol{x}, t)$ in the melt, where $\boldsymbol{x}$ is position and $t$ time, in terms of the following material properties: solid density $\rho_{s}$, specific heat capacity $C_{p}$, coefficient of latent heat per unit mass $L$, thermal conductivity $k$, melting temperature $\theta_{m}$. You may assume that the densities of the melt and the solid are the same and that temperature in the melt far from the interface is $\theta_{m}-\Delta \theta$, where $\Delta \theta$ is a positive constant.

A spherical crystal of radius $a(t)$ grows into such a melt with $a(0)=0$. Use dimensional analysis to show that $a(t)$ is proportional to $t^{1 / 2}$.

(ii) Show that the above problem should have a similarity solution of the form

$$\theta=\theta_{m}-\Delta \theta(1-F(\xi))$$

where $\xi=r(\kappa t)^{-1 / 2}, r$ is the radial coordinate in spherical polars and $\kappa=k / \rho_{s} C_{p}$ is the thermal diffusivity. Recalling that, for spherically symmetric $\theta, \nabla^{2} \theta=\frac{1}{r^{2}}\left(r^{2} \theta_{r}\right)_{r}$, write down the equation and boundary conditions to be satisfied by $F(\xi)$. Hence show that the radius of the crystal is given by $a(t)=\lambda(\kappa t)^{1 / 2}$, where $\lambda$ satisfies the equation

$$\int_{\lambda}^{\infty} \frac{e^{-\frac{1}{4} u^{2}}}{u^{2}} d u=\frac{2}{S \lambda^{3}} e^{-\frac{1}{4} \lambda^{2}}$$

and $S=L / C_{p} \Delta \theta$.

Integrate the left hand side of this equation by parts, to give

$$\frac{\sqrt{\pi}}{2} \lambda e^{\frac{1}{4} \lambda^{2}} \operatorname{erfc}\left(\frac{1}{2} \lambda\right)=1-\frac{2}{S \lambda^{2}} .$$

Hence show that a solution with $\lambda$ small must have $\lambda \approx(2 / S)^{\frac{1}{2}}$, which is self-consistent if $S$ is large.