= Solution
In the narrowest part of the film, $H=O(1)$, so $h\sim s=2Q/U$. Therefore
$$
\boxed{C=\frac{s}{a}\ll1}
$$
is exactly the thin-annulus requirement needed to replace the local circumference by $2\pi a$ and neglect curvature across the film. The axial lubrication scale is $\ell=\sqrt{s/\kappa}$, so the small-slope assumption additionally requires $s/\ell=\sqrt{\kappa s}\ll1$. The assumed negligible inertia imposes its corresponding small inertial-to-viscous ratio, of order $\rho Us^2/(\mu\ell)\ll1$. Thus small $C$ is necessary, while slope and inertia conditions are independent parts of self-consistency. If the lubrication approximation is required all the way to large scaled endpoints, the leading parabolic gap further requires $C\widetilde L^2\ll1$ and $\sqrt{\kappa s}\,\widetilde L\ll1$. Alternatively the thin neck can be matched to a separate outer solution; its <pressure> contribution must then be assessed.
Back to article page
