Linear instability of a two-dimensional viscoelastic liquid film under the coupling effect of electrostatic field and parametric resonance

A two-dimensional temporal linear instability analysis is performed for a charged liquid viscoelastic film on a vertically oscillating plane in the presence of an electrostatic field. The viscoelastic fluid, described by the Oldroyd-B model, is intended to be a Taylor-Melcher leaky dielectric, while the ambient gas is treated as perfectly dielectric. Results show that the oscillations induce parametric unstable regions. The parametric unstable regions can be superimposed on the inherent unstable region when the oscillation frequency is small because the frequency in the inherent unstable region is non-zero. The electric field has a complex effect on both inherent and parametric instabilities. The effect of the electrical relaxation time is dominated by the electrical Bond number and dimensionless distance. The relative permittivity has a destabilizing effect. However, the effects of electrical relaxation time and relative permittivity in the parametric unstable region are negligible. The viscosity and elasticity have non-monotonic effects on the inherent instability but have a monotonous effect on the parametric instability. The increase in Bond number has a stabilizing effect. The effect of density ratio is not constant in the inherent unstable region according to the electrical Bond number and Bond number. However, the increase of density ratio has a monotonously stabilizing effect on the parametric instability.