Dynamics and stability of leaky dielectric viscous films coating the inside of a vertical cylindrical tube in a radial electric field

The dynamics of an electrified viscous film coating the interior of a vertical cylindrical tube subject to a radial electrostatic field and driven by gravity is investigated theoretically. We consider the general case where both the inner gas and the annular liquid film are weakly conductive with different electrical properties. Using the Taylor-Melcher leaky dielectric model, a coupled system of evolution equations governing the interfacial position and charge distribution is derived for the axisymmetric flow with long-wave (LW) asymptotics. Linear stability analysis of the electrified film to axisymmetric disturbances is carried out. The influences of various parameters on stability characteristics have been explored in detail, including the radii of outer and inner electrodes, relative permittivities, dimensionless conductivities Σ_i, an electric Weber number E_w, and a Reynolds number. The electric field is found to be either stabilizing or destabilizing; and in the limiting case of high Σ_i in both phases, the stability conditions for the dual role of the electric field can be deduced by simply analyzing a normal-stress balance at the gas-liquid interface. In leaky dielectric cases, the dispersion curves have two branches if Σ_i was not too large and E_w is sufficiently high, in contrast to perfectly dielectric cases where only one branch is obtained. In the nonlinear regime, the temporal evolution of the instabilities is predicted by solving the LW model numerically in a wide range of parameters. Rich dynamics are revealed and classified, including steady-state and time-periodic traveling waves, finite-time touchdown singularity, and complex chaotic dynamics. Phase diagrams of the flow behaviors are constructed in the parameter space with extensive numerical simulations.