(i) A layer of fluid of depth $h(x, t)$, density $\rho$ and viscosity $\mu$ sits on top of a rigid horizontal plane at $y=0$. Gravity $g$ acts vertically and surface tension is negligible.

Assuming that the horizontal velocity component $u$ and pressure $p$ satisfy the lubrication equations

$$\begin{aligned} &0=-p_{x}+\mu u_{y y} \\ &0=-p_{y}-\rho g, \end{aligned}$$

together with appropriate boundary conditions at $y=0$ and $y=h$ (which should be stated), show that $h$ satisfies the partial differential equation

$$h_{t}=\frac{g}{3 \nu}\left(h^{3} h_{x}\right)_{x},$$

where $\nu=\mu / \rho$.

(ii) A two-dimensional blob of the above fluid has fixed area $A$ and time-varying width $2 X(t)$, such that

$$A=\int_{-X(t)}^{X(t)} h(x, t) d x$$

The blob spreads under gravity.

Use scaling arguments to show that, after an initial transient, $X(t)$ is proportional to $t^{1 / 5}$ and $h(0, t)$ is proportional to $t^{-1 / 5}$. Hence show that equation $(*)$ of Part (i) has a similarity solution of the form

$$h(x, t)=\left(\frac{A^{2} \nu}{g t}\right)^{1 / 5} H(\xi), \quad \text { where } \quad \xi=\frac{x}{\left(A^{3} g t / \nu\right)^{1 / 5}}$$

and find the differential equation satisfied by $H(\xi)$.

Deduce that

$$H= \begin{cases}{\left[\frac{9}{10}\left(\xi_{0}^{2}-\xi^{2}\right)\right]^{1 / 3}} & \text { in }-\xi_{0}<\xi<\xi_{0} \\ 0 & \text { in }|\xi|>\xi_{0}\end{cases}$$

where

$$X(t)=\xi_{0}\left(\frac{A^{3} g t}{\nu}\right)^{1 / 5}$$

Express $\xi_{0}$ in terms of the integral

$$I=\int_{-1}^{1}\left(1-u^{2}\right)^{1 / 3} d u$$