Consider a two-dimensional flow of a viscous fluid down a plane inclined at an angle $\alpha$ to the horizontal. Initially, the fluid, which has a volume $V$, occupies a region $0 \leqslant x \leqslant x^{*}$ with $x$ increasing down the slope. At large times the flow becomes thin-layer flow.

(i) Write down the two-dimensional Navier-Stokes equations and simplify them using the lubrication approximation. Show that the governing equation for the height of the film, $h=h(x, t)$, is

$$\tag{†} \frac{\partial h}{\partial t}+\frac{\partial}{\partial x}\left(\frac{g h^{3} \sin \alpha}{3 \nu}\right)=0$$

where $\nu$ is the kinematic viscosity of the fluid and $g$ is the acceleration due to gravity, being careful to justify why the streamwise pressure gradient has been ignored compared to the gravitational body force.

(ii) Develop a similarity solution to $(†)$ and, using the fact that the volume of fluid is conserved over time, derive an expression for the position and height of the head of the current downstream.

(iii) Fluid is now continuously supplied at $x=0$. By using scaling analysis, estimate the rate at which fluid would have to be supplied for the head height to asymptote to a constant value at large times.