Electrohydrodynamics of leaky dielectric falling films coating the interior of a tubular electrode: linear stability and nonlinear interfacial dynamics

We formulated an electrostatically modified Orr–Sommerfeld eigenvalue problem using the Taylor–Melcher leaky dielectric model. This eigenvalue problem has been solved with the compound matrix method based on a generalization of the Evans function. The numerical results of electrified Stokes problems predict that apart from the classic Plateau–Rayleigh instability in the long-wave (LW) range, surface wave (SW) and leaky-dielectric (LD) modes may appear. For a given outer electrode radius (α) and a relatively smaller inner electrode radius (β), LD mode could occur in two branches of dispersion curves with SW modes in two disjoint intervals of smaller and larger wavenumbers (k). Neutral curves in the β−k plane are calculated for different values of α, the relative permittivity of liquid, and an electric Weber number. Next, we derived coupled LW evolution equations for the axisymmetrically interfacial position and charge distribution. Our parametric study shows that only in a narrow window of relatively small values of permittivity ratio can traveling waves occur. It is shown that either outer- or inner-electrode finite-time touchdown (TD) is possible for leaky dielectric cases, while only inner electrode TD is found for a pair of perfect dielectric liquids. The results demonstrate that rupture behaviors can be controlled by changing electric parameters. Phase diagrams show the existence of a large stable equilibrium region whose size increases with the dimensionless conductivities.