= Solution
Let the cylinder radius be $a$, with $h/a\ll1$ and slowly varying thickness. Arc length is $s=a\theta$, with increasing $\theta$ following the upward-moving surface at the right-hand side. Gravity has tangential component $-g\cos\theta$. In the leading thin-film balance, normal viscous gradients dominate tangential ones, curvature corrections are small, and the longitudinal hydrostatic-pressure gradient is smaller by the aspect ratio. Neglect capillary-pressure gradients and imposed gas shear. The tangential Stokes balance is then
$$
\boxed{\nu\,\partial_z^2u=g\cos\theta.}
$$
These assumptions also require that inertia and curvature effects from rotation remain small on the thin-film scale; an arbitrary high-speed coating need not obey this equation.
No-slip at the solid wall and negligible <shear stress> at the <free surface> give
$$
u(\theta,0)=U,\qquad \left.\partial_zu\right|_{z=h}=0.
$$
Twice integrating the constant normal forcing determines the two integration constants:
$$
\boxed{u(\theta,z)=U+\frac g\nu\left(\frac{z^2}{2}-zh\right)\cos\theta.}
$$
The profile is most strongly retarded by gravity on the ascending side. The surface <velocity> is $U-gh^2\cos\theta/(2\nu)$, not generally equal to the wall speed.
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