= Solution
Let the supplied volume per unit width be $V(t)$. The governing balance gives the scales
$$
\frac ht\sim C\frac{h^3}{x},
\qquad
V\sim hx.
$$
Eliminating $x$ yields
$$
V(t)\sim Ch^3t.
$$
For the head height to approach a nonzero constant $h_\infty$, the total supplied volume must therefore grow linearly:
$$
\boxed{
\frac{dV}{dt}\sim C h_\infty^3
=\frac{g\sin\alpha}{3\nu}h_\infty^3
=\text{constant}
}.
$$
This is the <constant-head supply for a gravity-driven thin film>.
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