 Solution

In the narrowest part of the film, $H=O(1)$, so $h\sim s=2Q/U$. Thereforeis 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 $s/\ell =\sqrt{\kappa s}\ll 1$. The assumed negligible inertia imposes its corresponding small inertial-to-viscous ratio, of order $\rho U{s}^2/(\mu \ell )\ll 1$. 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 $C{\tilde{L}}^2\ll 1$ and $\sqrt{\kappa s}\tilde{L}\ll 1$. Alternatively the thin neck can be matched to a separate outer solution; its pressure contribution must then be assessed.