(i) Material of thermal diffusivity $D$ occupies the semi-infinite region $x>0$ and is initially at uniform temperature $T_{0}$. For time $t>0$ the temperature at $x=0$ is held at a constant value $T_{1}>T_{0}$. Given that the temperature $T(x, t)$ in $x>0$ satisfies the diffusion equation $T_{t}=D T_{x x}$, write down the equation and the boundary and initial conditions satisfied by the dimensionless temperature $\theta=\left(T-T_{0}\right) /\left(T_{1}-T_{0}\right)$.

Use dimensional analysis to show that the lengthscale of the region in which $T$ is significantly different from $T_{0}$ is proportional to $(D t)^{1 / 2}$. Hence show that this problem has a similarity solution

$$\theta=\operatorname{erfc}(\xi / 2) \equiv \frac{2}{\sqrt{\pi}} \int_{\xi / 2}^{\infty} e^{-u^{2}} d u$$

where $\xi=x /(D t)^{1 / 2}$.

What is the rate of heat input, $-D T_{x}$, across the plane $x=0 ?$

(ii) Consider the same problem as in Part (i) except that the boundary condition at $x=0$ is replaced by one of constant rate of heat input $Q$. Show that $\theta(\xi, t)$ satisfies the partial differential equation

$$\theta_{\xi \xi}+\frac{\xi}{2} \theta_{\xi}=t \theta_{t}$$

and write down the boundary conditions on $\theta(\xi, t)$. Deduce that the problem has a similarity solution of the form

$$\theta=\frac{Q(t / D)^{1 / 2}}{T_{1}-T_{0}} f(\xi)$$

Derive the ordinary differential equation and boundary conditions satisfied by $f(\xi)$.

Differentiate this equation once to obtain

$$f^{\prime \prime \prime}+\frac{\xi}{2} f^{\prime \prime}=0$$

and solve for $f^{\prime}(\xi)$. Hence show that

$$f(\xi)=\frac{2}{\sqrt{\pi}} e^{-\xi^{2} / 4}-\xi \operatorname{erfc}(\xi / 2)$$

Sketch the temperature distribution $T(x, t)$ for various times $t$, and calculate $T(0, t)$ explicitly.