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The theoretical and numerical models for gravity driven coating flow on uppercylinder and sphere are formulated. Using a perturbation method, the governingequations which depend on one Bond number $Bo$ are derived for a liquid filmflow down the outside of a horizontal cylinder and a sphere. They can besimplified to one-dimensional form due to the symmetries. The general structureof the two-dimensional and axisymmetric capillary waves under high $Bo$condition is focused on. An asymptotic theory is used to solve for thefree-surface profiles in the outer and inner region, respectively. Even thoughthe evolution in the outer region is essentially different, there are inherentsimilarities in the inner region because the capillary ridges are proved todegenerate into the one-parameter family in high $Bo$ limit. Using appropriatenumerical techniques, some parametric studies are performed on the profiles ofthe capillary waves. The spreading of the front recorded from the outersolutions quantitatively accords with the scaling laws in a top region at earlytime, but deviates obviously at later time. The comparison between thecomposite solutions constructed using the asymptotic theory and directsolutions calculated from the evolution equations is highlighted via both theglobal and local features. Agreement between the composite solutions and thedirect solutions is good for high $Bo$ cases. This asymptotic behavior iscommon on both of cylindrical and spherical surfaces and not affected by thepartial wetting process.

A theoretical and numerical study of gravity driven coating flow on cylinder and sphere: two-dimensional and axisymmetric capillary waves