Solution
= Solution
The <volume flux per unit width> is
$$
Q=\int_0^hu(y)\,dy
=\boxed{\frac{\rho gh^3}{12\mu}
\frac{4+kh/\mu}{1+kh/\mu}.}
$$
Therefore
$$
\boxed{Q\longrightarrow\frac{\rho gh^3}{3\mu}\quad(k\to0),\qquad
Q\longrightarrow\frac{\rho gh^3}{12\mu}\quad(k\to\infty).}
$$
The first is the stress-free film; the second has an effectively immobilized free surface.