Modal and nonmodal stability analysis for an electrified falling film.

Abstract

We have performed the modal and nonmodal stability analyses of a gravity-driven three-dimensional viscous incompressible fluid flowing over an inclined plane in the presence of a uniform electric field acting normal to the plane at infinity. The time evolution equationsare derived for normal velocity, normal vorticity, and fluid surface deformation, respectively, and solved numerically by using the Chebyshev spectral collocation method. The modal stability analysis demonstrates the existence of three unstable regions for the surface mode in the wave number plane at the lower value of the electric Weber number. However, these unstable regions coalesce and magnify as the electric Weber number rises. By contrast, there exists only one unstable region for the shear mode in the wave number plane, which attenuates slightly with an increase in the value of the electric Weber number. But both the surface and shear modes are stabilized in the presence of the spanwise wave number, where the long-wave instability shifts towards the finite wavelength instability as the spanwise wave number rises. On the other hand, the nonmodal stability analysis reveals the existence of transient disturbance energy growth, the maximum value of which intensifies slightly with an increase in the value of the electric Weber number.