Modeling and analysis of two electrified films flow traveling down between inclined permeable parallel substrates

摘要

The purpose of this study is to establish the temporal stability of two bounded thin films flow of a viscous fluid inside a permeable inclined channel. Based on the long-wave theory, an integral boundary layer model for the film thickness, the volumetric flow rate, and the surface charge are derived. The driving force for the instability under an electric field is an electrostatic force exerted on the free charges accumulated at the interface. The linear stability analysis for the leaky dielectric model is performed, and a cubic dispersion relation is obtained by the normal mode technique using suitable boundary and interface conditions. The numerical calculations of the linear analysis reveal that our model is unstable for a small Reynolds number, and for higher numbers, the system becomes stable in nature. The dielectric constant ratio has a stabilizing influence, in which the inverse behavior is found for increasing the electrical conductivity. For the perfect dielectric case, the nonlinear stability is carried out. The analytical solution of stationary waves is discussed by introducing the linearized instability of the fixed points and Hopf bifurcation. It is found that the viscosity ratio and the permeability parameter have an opposite effect on the existence of the fixed points. A specified case of the stationary wave, namely Shkadov wave, is investigated.