Matching the flat film to a cylindrical interface gives the local <parabolic lubrication gap>
$$
h-h_0\sim\frac{(x-L)^2}{a}.
$$
The thickness changes by $O(h_0)$ across the transition, so
$$
\boxed{\delta\sim(ah_0)^{1/2}}.
$$
In the extensional-force equation, viscous and capillary terms have scales
$$
\frac{\mu h_0U}{\delta^2},
\qquad
\frac{\gamma h_0^2}{\delta^3}.
$$
Their balance gives
$$
\boxed{U\sim\frac\gamma\mu\left(\frac{h_0}{a}\right)^{1/2}}.
$$

The transition adjusts on time $\delta/U$, whereas the flat film drains on time $L/U$. Since $\delta\ll L$, the transition is quasi-steady. Its <thin-film mass flux> therefore satisfies
$$
\boxed{uh\simeq Uh_0=\text{constant}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 329 / 2 / c / Solution

The thin-film mass flux is the layer thickness times its depth-averaged velocity. In a quasi-steady transition region, mass conservation makes this flux approximately constant along the layer.


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