Numerical study for the electrified instability of viscoelastic cylindrical dielectric fluid film surrounded by a conducting gas

Abstract

The linear electrohydrodynamic cylindrical instability of annular Walters B ′ viscoelastic dielectric fluid layer surrounded by a conducting gas in the presence of radial electric field is investigated. The obtained dispersion relation is found to be complicated and cannot be treated theoretically easily. Two limiting cases of interest are investigated, when the inertia is dominant, and when both the kinematic viscosity and viscoelasticity are high, and the corresponding new stability conditions are obtained for both cases. We solve the eigenvalue problem numerically using the continuation method which gives better results than the classical non-linear solvers such as Newton and Secant methods. It is found that the applied radial electric field has a dual role on the stability of the considered system, depending of the chosen wavenumbers range. Both the kinematic viscoelasticity and liquid depth are found to have stabilizing effects, while both the kinematic viscosity and surface tension have destabilizing effects on the considered system. The stability or instability breaks down for critical wavenumber values at which the growth rate vanishes. The behaviors of both the maximum growth rate and the corresponding dominant wavenumber are discussed in detail corresponding to the effect of all physical parameters. Finally a comparison between the results obtained here for Walters B ′ viscoelastic fluids, and those obtained here too if the fluid is replaced by a Rivlin–Ericksen viscoelastic one is achieved. The limiting cases of absence of electric field and/or kinematic viscoelasticity are also investigated in detail.