= 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 $\rho U b^2/(\mu\ell)\ll1$. 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.
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