# Two-dimensional pulse dynamics and the formation of bound states on electrified falling films

## TL;DR

This paper investigates the dynamics of electrified falling liquid films, focusing on solitary wave structures, their interactions, and how electric fields influence stability, amplitude, and bound states, using theoretical models and numerical simulations.

## Contribution

It introduces a combined analysis of long-wave and Stokes models to understand electric field effects on pulse formation, stability, and bound states in falling films.

## Key findings

- Electric fields increase pulse amplitude and induce recirculation zones.
- Bound states of pulses exist and are influenced by electric field strength.
- Electric fields can switch flow stability from absolute to convective.

## Abstract

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destablising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical elec- tric field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps, and alters the far-field decay of the pulse tails from ex- ponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute instability to convective instability, thereby regularising the dynamics, and this is confirmed by time- dependent simulations of the long-wave model.