= Solution
Balancing the downslope gravitational force density $\rho g\sin\alpha$ against the viscous term $\mu u_{yy}$ gives the velocity scale
$$
\boxed{U\sim\frac{\rho g h^2\sin\alpha}{\mu}}.
$$
If the streamwise variation scale is $L\sim h/|h_x|$, the <lubrication theory> requirement is first
$$
\boxed{|h_x|\ll1}.
$$
The ratio of streamwise inertia $\rho U^2/L$ to transverse viscous stress $\mu U/h^2$ is
$$
\frac{\rho Uh^2}{\mu L}
\sim
\frac{\rho^2g h^3\sin\alpha}{\mu^2}|h_x|.
$$
Thus self-consistency also requires
$$
\boxed{
\frac{\rho^2g h^3\sin\alpha}{\mu^2}|h_x|\ll1}.
$$
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