= Solution
Balance the two terms in the <planar viscous-sheet stretching equations> over the transition. The <velocity> changes by $O(U)$ on length $\delta$, and $h_{xxx}=O(h_0/\delta^3)=O(1/(a\delta))$. Thus
$$
\frac{\mu h_0U}{\delta^2}\sim\frac{\gamma h_0}{a\delta}.
$$
Using the geometric transition length gives
$$
\boxed{U\sim\frac\gamma\mu\frac\delta a
\sim\frac\gamma\mu\sqrt{\frac{h_0}a}.}
$$
This balances capillary forcing with planar extensional <viscous stress>, not the no-slip Poiseuille resistance appropriate to a film bounded by rigid walls.
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