Discuss how the methods of lubrication theory may be used to find viscous fluid flows in thin layers or narrow gaps, explaining carefully what inequalities need to hold in order that the theory may apply.

Viscous fluid of kinematic viscosity $\nu$ flows under the influence of gravity $g$, down an inclined plane making an angle $\alpha \ll 1$ with the horizontal. The fluid layer lies between $y=0$ and $y=h(x, t)$, where $x, y$ are distances measured down the plane and perpendicular to it, and $|\partial h / \partial x|$ is of the same order as $\alpha$. Give conditions involving $h, \alpha, \nu$ and $g$ that ensure that lubrication theory can be used, and solve the lubrication equations, together with the equation of mass conservation, to obtain an equation for $h$ in the form

$$\frac{\partial h}{\partial t}=\frac{\partial}{\partial x}\left(-A h^{3}+B h^{3} \frac{\partial h}{\partial x}\right)$$

where $A, B$ are constants to be determined. Show that there is a steady solution with $\partial h / \partial x=k=$ constant, and interpret this physically. Show also that a solution of this equation exists in the form of a front, with $h(x, t)=F(\xi)$, where $\xi=x-c t, F(0)=0$, and $F(\xi) \rightarrow h_{0}$ as $\xi \rightarrow-\infty$. Determine $c$ in terms of $h_{0}$, find the shape of the front implicitly in the form $\xi=G(h)$, and show that $h \propto(-\xi)^{1 / 3}$ as $\xi \rightarrow 0$ from below.