Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 37E / Solution

Lubrication theory exploits a film thickness $b$ much smaller than its horizontal variation scale $\ell$. Incompressibility then gives vertical velocity of order $(b/\ell )U$, vertical viscous derivatives dominate horizontal ones, and inertia is negligible when $\rho U{b}^2/(\mu \ell )\ll 1$. The leading equations reduce to $\mu u_{yy}=p_x$ and vertical hydrostatic balance $p_y=-\rho g$. Boundary conditions determine the local horizontal velocity profile; integrating it across the film and enforcing mass conservation produces an equation for flux or surface shape. This reduction turns the thin-film problem into a one-dimensional pressure problem while retaining the dominant viscous stresses.