= Edge region of a draining viscous film
The outer <similarity solution> approaches a plate edge as $h/h_0\sim[(4C/5)(1-x/L)]^{1/4}$, where $C=B(4/5,1/2)$. Its slope becomes order one at
$$
\ell\sim L(h_0/L)^{4/3}.
$$
This region lies outside <lubrication theory> because its vertical and horizontal scales are comparable. The outer zero-thickness boundary condition is consistent at leading order since the matching height is $O(\ell)\ll h_0$, while the actual edge flow requires an inner solution. A finite outward flux survives even though the outer thickness vanishes, because the slope diverges.
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