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