Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization

TL;DR

This paper develops a rigorous theory for the interaction of solitary pulses on falling liquid films, highlighting how viscous dispersion influences bound-state formation and self-organization, with predictions matching detailed model computations.

Contribution

The paper introduces a coherent-structure theory that predicts bound-state distances in falling liquid films considering viscous dispersion effects, advancing understanding of self-organization in such systems.

Findings

Viscous dispersion affects bound-state distances.

The theory accurately predicts pulse separation distances.

Bound states lead to a self-organized wave gas with characteristic density.

Abstract

We examine the interaction of two-dimensional solitary pulses on falling liquid films. We make use of the second-order model derived by Ruyer-Quil and Manneville [Eur. Phys. J. B 6, 277 (1998); Eur. Phys. J. B 15, 357 (2000); Phys. Fluids 14, 170 (2002)] by combining the long-wave approximation with a weighted residuals technique. The model includes (second-order) viscous dispersion effects which originate from the streamwise momentum equation and tangential stress balance. These effects play a dispersive role that primarily influences the shape of the capillary ripples in front of the solitary pulses. We show that different physical parameters, such as surface tension and viscosity, play a crucial role in the interaction between solitary pulses giving rise eventually to the formation of bound states consisting of two or more pulses separated by well-defined distances and travelling at the same velocity. By developing a rigorous coherent-structure theory, we are able to theoretically predict the pulse-separation distances for which bound states are formed. Viscous dispersion affects the distances at which bound states are observed. We show that the theory is in very good agreement with computations of the second-order model. We also demonstrate that the presence of bound states allows the film free surface to reach a self-organized state that can be statistically described in terms of a gas of solitary waves separated by a typical mean distance and characterized by a typical density.