Dynamics of an electrostatically modified Kuramoto–Sivashinsky–Korteweg–de Vries equation arising in falling film flows

Abstract

The gravity-driven flow of a liquid film down a vertical flat plate in the presence of an electric field acting in a direction perpendicular to the wall is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. A strongly nonlinear long-wave evolution equation is developed for flow parameters near the critical instability conditions. The equation retains terms up to second order in the slenderness parameter in order to incorporate the effects of shear-induced growth, short-wave damping due to surface tension, electric stress effects, and dispersive effects. In the additional asymptotic limit of small but finite interfacial perturbations the dynamics are shown to be governed by a Kuramoto-Sivashinsky (KS) equation with Korteweg-de Vries dispersion, also known as the Kawahara or the generalized Kuramoto-Sivashinsky (gKS) equation, which also includes a nonlocal energy growth term that arises from the electrostatics. Extensive numerical experiments are carried out to characterize solutions to this equation. Using perturbation theory and numerical solutions, it is shown that the electric field alters the far-field decay characteristics of bound states of the gKS equation from exponential to algebraic behavior. In addition, it is demonstrated numerically that chaotic solutions of the KS equation that are regularized into traveling-wave pulses when sufficient dispersion is added can in turn become chaotic by applying a sufficiently strong electric field. It is suggested, therefore, that electric fields can be utilized to enhance interfacial turbulence and in turn increase heat or mass transfer in applications. A physical example involving electrified falling film flows of ethelene glycol fluids is furnished and shows that the theory is within reach of experiments.