= Solution
The volume flux per unit transverse width is
$$
\begin{aligned}
q(x,t)
&=\int_0^h u\,dy\\
&=\frac{\rho gh^3}{3\mu}
\left(\sin\alpha-\cos\alpha\,h_x\right).
\end{aligned}
$$
Integrating incompressibility across the layer and using the wall and free-surface kinematic conditions gives local mass conservation $h_t+q_x=0$. Hence the <gravity-driven thin film on an incline> obeys
$$
\boxed{
h_t+\frac{\rho g}{3\mu}
\partial_x\left[
h^3(\sin\alpha-\cos\alpha\,h_x)
\right]=0}.
$$
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