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To find the regularities of formed fingers in gravity driven coating flows onupper cylinder and sphere, the mathematical formulation to model the fingeringinstability on cylindrical or spherical surface which consists of a capillarywave equation and a linear perturbation equation is constructed, based on theleading order governing equation in standard cylindrical or sphericalcoordinate system. A disjoining pressure model is introduced to simulate thepartial wetting process near moving contact line. The fingering number in high$Bo$ coating flow is focused on from a linear perspective. Using an asymptotictheory, the high $Bo$ limits of the linear perturbation equations for both thecylindrical and spherical problems are proved to degenerate into a commoneigenvalue problem. Two analytical formulae concerning the most unstable wavenumber which can be described as two power laws are derived via a method ofmodal analysis for the cylindrical and spherical problem, respectively. Theseformulae provide a succinct method to estimate the fingering number if theonset of the fingering instability is known. A method of transient growthanalysis which makes no high $Bo$ assumptions is used to verify the accuracy ofthe asymptotic theory and modal analysis. In particular, an asymptotic behaviorof the most unstable wave number as $Bo$ increases is highlighted. We show thatthe linear growth rate may be improved by the disjoining pressure, which can beattributed to the effect of partial wetting, but the most unstable wave numberis insensitive to the wetting model.

A theoretical and numerical study of gravity driven coating flow on cylinder and sphere: fingering instability