  The theoretical and numerical models for gravity driven coating flow on upper
cylinder and sphere are formulated. Using a perturbation method, the governing
equations which depend on one Bond number $Bo$ are derived for a liquid film
flow down the outside of a horizontal cylinder and a sphere. They can be
simplified to one-dimensional form due to the symmetries. The general structure
of the two-dimensional and axisymmetric capillary waves under high $Bo$
condition is focused on. An asymptotic theory is used to solve for the
free-surface profiles in the outer and inner region, respectively. Even though
the evolution in the outer region is essentially different, there are inherent
similarities in the inner region because the capillary ridges are proved to
degenerate into the one-parameter family in high $Bo$ limit. Using appropriate
numerical techniques, some parametric studies are performed on the profiles of
the capillary waves. The spreading of the front recorded from the outer
solutions quantitatively accords with the scaling laws in a top region at early
time, but deviates obviously at later time. The comparison between the
composite solutions constructed using the asymptotic theory and direct
solutions calculated from the evolution equations is highlighted via both the
global and local features. Agreement between the composite solutions and the
direct solutions is good for high $Bo$ cases. This asymptotic behavior is
common on both of cylindrical and spherical surfaces and not affected by the
partial wetting process.
