  We derive a generalised asymptotic model for the flow of a thin fluid film
over an arbitrarily-parameterised non-axisymmetric curved substrate surface
based on the lubrication approximation. In addition to surface tension,
gravity, and centrifugal force, our model incorporates the effects of the
Coriolis force and disjoining pressure, together with a non-uniform initial
condition, which have not been widely considered in existing literature. We use
this model to investigate the impact of the Coriolis force and fingering
instability on the spreading of a non-axisymmetric spin-coated film at a range
of substrate angular velocities, first on a flat substrate, and then on
parabolic cylinder- and saddle-shaped curved substrates. We show that, on flat
substrates, the Coriolis force has a negligible impact at low angular
velocities, and at high angular velocities results in a small deflection of
fingers formed at the contact line against the direction of substrate rotation.
On curved substrates, we demonstrate that as the angular velocity is increased,
spin coated films transition from being dominated by gravitational drainage
with no fingering to spreading and fingering in the direction with the greatest
component of centrifugal force tangent to the substrate surface. For both
curved substrates and all angular velocities considered, we show that the film
thickness and total wetted substrate area remain similar over time to those on
a flat substrate, with the key difference being the shape of the spreading
droplet.
