  The stability and axisymmetric deformation of two immiscible, viscous,
perfect or leaky dielectric fluids confined in the annulus between two
concentric cylinders are studied in the presence of radial electric fields. The
fields are set up by imposing a constant voltage potential difference between
the inner and outer cylinders. We derive a set of equations for the interface
in the long-wavelength approximation which retains the essential physics of the
system and allows for interfacial deformations to be as large as the annular
gap hence accounting for possible touchdown at the inner or outer electrode.
The effects of the electric parameters are evaluated initially by performing a
linear stability analysis which shows excellent agreement with the linear
theory of the full axisymmetric problem in the appropriate long wavelength
regime. The nonlinear interfacial dynamics are investigated by carrying out
direct numerical simulations of the derived long wave models, both in the
absence and presence of electric fields. For non-electrified thin layer flows
(i.e. one of the layers thin relative to the other) the long-time dynamics
agree with the lubrication approximation results found in literature. When the
liquid layers have comparable thickness our results demonstrate the existence
of both finite time and infinite time singularities (asymptotic touching
solutions) in the system. It is shown that a two-side touching solution is
possible for both the non-electrified and perfect dielectric cases, while only
one-side touching is found in the case of leaky dielectric liquids, where the
flattened interface shape resembles the pattern solutions found in literature.
Meanwhile the finite-time singular solution agrees qualitatively with the
experiments of~\cite{Reynolds1965}.
