Effect of electric field on the linear stability of an inclined oscillatory film flow

The linear stability of a liquid film flowing down an oscillating inclined plane with an electric field acting normal to the plane is found for disturbances of arbitrary wavenumbers. Utilizing the long-wave perturbation method and Floquet theory, the analytical solutions of the time-dependent perturbation equations are solved. The asymptotic solution reveals that the Floquet exponent is independent of the wall-normal oscillation amplitude. The combined effects of the inclined angle, the electric field, and the oscillating amplitude on the stability of liquid film are described. Moreover, the physical mechanism of the destabilized effect of electric field in the long-wave regime is discussed briefly. For finite-wavelength instability, the time-dependent Floquet system is resolved numerically based on the Chebyshev spectral collocation. Both the effects of wall-normal oscillation and streamwise oscillation on the stability are considered. The results of wall-normal oscillation show that the presence of an electric field increases the critical amplitude and the cutoff wavenumber for the regions of subharmonic and harmonic instabilities, and the gravitational instability extends to larger wavenumbers. For the streamwise oscillation, a new island-like unstable region is observed in the contour of growth rate, leading to the breaking of the neutral curve. When considering the effect of electric field, the gravitational unstable region continuously expand to lower oscillation frequencies, and the stable region only exists within closed neutral curves. This demonstrates the destabilizing effect of electric field on oscillating liquid film.