  Although the drainage and spreading processes of thin liquid films on
substrates have received growing attention during the last decades, the study
of three-dimensional cases is limited to a few studies on flat and axisymmetric
substrates. In this work, we exploit differential geometry to study the
drainage and spreading of thin films on generic curved substrates. We initially
investigate the drainage and spreading processes on spheroidal and paraboloidal
substrates by employing an asymptotic expansion in the vicinity of the pole and
a self-similar approach, finding that the thickness distribution is set by the
substrate metric and tangential gravity force components. Spheroids with a
large ratio between height and equatorial radius are characterized by a growing
thickness moving away from the pole, and vice versa. The non-symmetric coating
on a toroidal substrate shows that larger thicknesses and a faster spreading
are attained on the inner region than on the outer region of the torus. An
ellipsoid with three different axes is chosen as a testing ground for
three-dimensional drainage and spreading. Modulations in the drainage solution
are observed, with a different variation of the thickness along the two axes.
By imposing the conservation of mass, an analytical solution for the average
spreading front is obtained. The analytical and numerical results are in good
agreement. The resulting drainage solutions show also a good agreement with
experimental measurements obtained from the coating of a curing polymer on
diverse substrates.
