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

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 ͑1998͒; Eur. Phys. J. B 15, 357 ͑2000͒; Phys. Fluids 14, 170 ͑2002͔͒ by combining the long-wave approximation with a weighted residual technique. The model includes ͑second-order͒ 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 traveling 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.

Self-similarity of solitary waves on inertia-dominated falling liquid films

“…height of the trough preceding the solitary wave) is independent of β and σ for high Reynolds numbers. Similarly, the influence of the inclination angle and the surface tension coefficient on the number and amplitude of capillary ripples preceding the main solitary hump reduces for increasing Reynolds number, which is particularly relevant for the onset of flow separation [4,[37][38][39] as well as the wave length and binary interactions of solitary waves [12,13]. A.…”

“…It is then essential that in order to understand * f.denner09@imperial.ac.uk the spatiotemporal evolution of the film, we fully understand the properties of individual solitary waves which in turn can help us understand the way they interact with each other. In fact, these coherent structures are truly elementary processes so that (by using techniques from nonlinear dynamics and dynamical systems theory) the dynamics of the film can be described by their superposition [9][10][11][12][13][14]. As a matter of fact the evolution of the film appears the result of interaction between solitarylike coherent structures which are stable and robust and interact indefinitely with each other as "quasi-particles".…”

Selection of solitary waves in vertically falling liquid films

“…Moreover, the preferred optimal solution among the three ones illustrated in Figure 3 is the one of intermediate frequency ( f = 5.2 Hz). This point was addressed in fact recently by Pradas et al [23] in the draggravity regime. Performing statistics on the separation lengths between the waves, they found peaks around the values corresponding to bound states predicted by the theory.…”

“…However, the selection process at work is weaker than in the drag-gravity regime (Figure 3d). Pradas et al [23] observed also that the pulses lose their boundstate self-organization in this regime. Their distribution functions of the mean-separation distance are quite broad but still show some peak.…”

“…When δ < 1.5, inertia effects are weak and the amplitude of the waves remains small (drag-gravity regime). Selection of the waves in this case has been the object of several works, within the framework of the hydrodynamic bound states theory, and based on various equation models: the weakly nonlinear Kawahara equation [16,22,32], the Benney equation [9], the Kapitza-Shkadov model [4] and more recently the Ruyer-Quil-Manneville model [23]. A discrete set of preferred wavelengths is observed as waves tend to form bound states through interaction of their tails with the capillary ripples at the front of the next ones.…”