  We consider the gravity-driven flow of a perfect dielectric, viscous, thin
liquid film, wetting a flat substrate inclined at a non-zero angle to the
horizontal. The dynamics of the thin film is influenced by an electric field
which is set up parallel to the substrate surface - this nonlocal physical
mechanism has a linearly stabilizing effect on the interfacial dynamics. Our
particular interest is in fluid films that are hanging from the underside of
the substrate; these films may drip depending on physical parameters, and we
investigate whether a sufficiently strong electric field can suppress such
nonlinear phenomena. For a non-electrified flow, it was observed by Brun et al.
(Phys. Fluids 27, 084107, 2015) that the thresholds of linear absolute
instability and dripping are reasonably close. In the present study, we
incorporate an electric field and analyse the absolute/convective instabilities
of a hierarchy of reduced-order models to predict the dripping limit in
parameter space. The spatial stability results for the reduced-order models are
verified by performing an impulse-response analysis with direct numerical
simulations (DNS) of the Navier-Stokes equations coupled to the appropriate
electrical equations. Guided by the results of the linear theory, we perform
DNS on extended domains with inflow/outflow conditions (mimicking an
experimental set-up) to investigate the dripping limit for both non-electrified
and electrified liquid films. For the latter, we find that the absolute
instability threshold provides an order-of-magnitude estimate for the electric
field strength required to suppress dripping; the linear theory may thus be
used to determine the feasibility of dripping suppression given a set of
geometrical, fluid and electrical parameters.
