Shock-waves between two electrified thin films flowing down nearly horizontal channel

The aim of this work is to study the dynamics of an electrified interface separating two immiscible layers in a nearly horizontal channel in the sense of shock waves. The long wave theory is utilized to derive an evolution equation for the interface position that models the two-dimensional flow in both co-and countercurrent configurations. This equation is parameterized by viscosity and density ratios, electric permittivity ratios, electric potential, and a total dimensionless flow rate; the system is further characterized by the height of the interface at the channel inlet and outlet, which are considered as extra parameters. In the present work, we have investigated the existence of various types of shock- like solutions, which include compressive Lax shocks, pairs of Lax and under-compressive shocks, and rarefaction waves. The surface wave structure is demonstrated in both linear and nonlinear models. In the linear case, we have studied the surface flow pattern whether the initial interface is flat or characterized by a smoothing step function. The evolution equation governs the initial interface varying spatially along the direction of streamwise as well as that of the nonlinear case are solved numerically. The numerical results have been employed to show the effects of the different physical parameters on the surface wave structure and the mechanism of instability of the different types of waves, such as spatio-temporal instability.