A thin layer of liquid of kinematic viscosity $\nu$ flows under the influence of gravity down a plane inclined at an angle $\alpha$ to the horizontal $(0 \leq \alpha \leq \pi / 2)$. With origin $O$ on the plane, and axes $O x$ down the line of steepest slope and $O y$ normal to the plane, the free surface is given by $y=h(x, t)$, where $|\partial h / \partial x| \ll 1$. The pressure distribution in the liquid may be assumed to be hydrostatic. Using the approximations of lubrication theory, show that

$$\frac{\partial h}{\partial t}=\frac{g}{3 \nu} \frac{\partial}{\partial x}\left\{h^{3}\left(\cos \alpha \frac{\partial h}{\partial x}-\sin \alpha\right)\right\} .$$

Now suppose that

$$h=h_{0}+\eta(x, t)$$

where

$$\eta(x, 0)=\eta_{0} e^{-x^{2} / a^{2}}$$

and $h_{0}, \eta_{0}$ and $a$ are constants with $\eta_{0} \ll a, h_{0}$. Show that, to leading order,

$$\eta(x, t)=\frac{a \eta_{0}}{\left(a^{2}+4 D t\right)^{1 / 2}} \exp \left\{-\frac{(x-U t)^{2}}{a^{2}+4 D t}\right\}$$

where $U$ and $D$ are constants to be determined.

Explain in physical terms the meaning of this solution.