Selection of solitary waves in vertically falling liquid films

Two-dimensional solitary waves at the surface of a film flow down a vertical plane are considered. When the system is subjected to inlet white noise, solitary waves are formed after an inception region and interact with each other. Using open-domain simulations of reduced equation models, we investigate numerically their late time process dynamics. Close to the instability threshold, the waves synchronize themselves into bound states. For higher values of the Reynolds number, the separation distance between the waves increases and the synchronization process at work is weaker. Performing statistics, we show that the mean characteristics of the waves correspond to the minimal value of the mean film thickness along the traveling-wave branch of solutions. In this regime, synchronization occurs through the waves tails which is associated with a change of scaling of the waves features. A similar behavior is observed performing simulations in periodic domains: the selected waves maximize the mean flow rate. (C) 2016 Elsevier Ltd. All rights reserved

Inertial effects on thin-film wave structures with imposed surface shear on an inclined plane

This study provides an extended approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio, epsilon. Approximations are made assuming a Reynolds number, Re ~ O(1/epsilon), and depth-averaging used to simplify the governing Navier-Stokes equations. A parallel Stokes flow is expected in the absence of any wave disturbance and a generalisation for the flow is based on a local quadratic profile. This approach provides a more general system which includes inertial effects and is solved numerically. Flow structures are compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are used to study film profile evolution. A parametric study is provided for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken and identifies new profiles not previously predicted when inertial effects are neglected

A multi-layer integral model for locally-heated thin film flow

Based on an approach used to model environmental flows such as rivers and estuaries, we develop a new multi-layered model for thin liquid film flow on a locally-heated inclined plane. The film is segmented into layers of equal thickness with the velocity and temperature of each governed by a momentum and energy equation integrated across each layer individually. Matching conditions applied between the layers ensure the continuity of down-plane velocity, temperature, stress and heat flux. Variation in surface tension of the liquid with temperature is considered so that local heating induces a surface shear stress which leads to variation in the film height profile (the Marangoni effect). Moderate inertia and heat convection effects are also included. In the absence of Marangoni effects, when the film height is uniform, we test the accuracy of the model by comparing it against a solution of the full heat equation using finite differences. The multi-layer model offers significant improvements over that of a single layer. Notably, with a sufficient number of layers, the solution does not exhibit local regions of negative temperature often predicted using a single-layer model. With Marangoni effects included the film height varies however we find heat convection can mitigate this variation by reducing the surface temperature gradient and hence the surface shear stress. Numerical results corresponding to the flow of water on a vertical plane show that very thin films are dominated by the Marangoni shear stress which can be sufficiently strong to overcome gravity leading to a recirculation in the velocity field. This effect reduces with increasing film thickness and the recirculation eventually disappears. In this case heating is confined entirely to the interior of the film leading to a uniform height profile

Pulse dynamics in a power-law falling film

We examine the stability, dynamics and interactions of solitary waves in a two-dimensional vertically falling thin liquid film that exhibits shear-thinning effects. We use a low-dimensional two-field model that describes the evolution of both the local flow rate and the film thickness and is consistent up to second-order terms in the long-wave expansion. The shear-thinning behaviour is modelled via a power-law formulation with a Newtonian plateau in the limit of small strain rates. our results show the emergence of a hysteresis behaviour as the control parameter (the Reynolds number) is increased which is directly related to the shear-thinning character of the liquid an can be quantified with both linear analysis arguments and a physical interpretation. We also study pulse interactions, observing that two pulses may attract or repel each other either monotonically or in an oscillatory manner. In large domains we find that for a given Reynolds number the final state depends on the initial condition, a consequence of the presence of multiple solutions

Three-dimensional instabilities of quasi-solitary waves in a falling liquid film

Abstract The stability of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\gamma _2$ travelling waves at the surface of a film flow down an inclined plane is considered experimentally and numerically. These waves are fast, one-humped and quasi-solitary. They undergo a three-dimensional secondary instability if the flow rate (or Reynolds number) is sufficiently high. Rugged or scallop wave patterns are generated by the interplay between a short-wave and a long-wave instability mode. The short-wave mode arises in the capillary region of the wave, with a mechanism of capillary origin which is similar to the Rayleigh–Plateau instability, whereas the long-wave mode deforms the entire wave and is triggered by a Rayleigh–Taylor instability. Rugged waves are observed at relatively small inclination angles. At larger angles, the long-wave mode predominates and scallop waves are observed. For a water film the transition between rugged and scallop waves occurs for an inclination angle around 12°

Characteristics of solitary waves on a falling liquid film sheared by a turbulent counter-current gas flow

We report an experimental investigation of a falling water film sheared by a turbulent counter-current air flow in an inclined rectangular channel. Film thickness and wave velocity measurements associated with visual observation are conducted to study the influence of the air flow on controlled traveling waves consisting of a large wave hump preceded by capillary ripples. First, we focus on the variation of the shape, amplitude and velocity of the waves as the gas velocity is gradually increased. We demonstrate that the amplitude of the main hump grows substantially even for moderate gas velocities, whereas modification of the wave celerity becomes significant above a specific gas velocity around 4 m/s, associated with an alteration of the capillary region. The influence of the gas flow on 3D secondary instabilities of the solitary waves detected in a previous study Kofman et al. (2014), namely rugged or scallop waves, is also investigated. We show that the capillary mode is damped while the inertial mode is enhanced by the interfacial shear. Next, the gas velocity is increased until the onset of upstream-moving patterns referred to as flooding in our experiments. At moderate inclination angles (typically < 7), flooding occurs for a gas velocity around 8 m/s and is initiated at the scallop wave crests by a backward wave-breaking phenomenon preceded by the onset of ripples on the flat residual film separating two waves. At high inclination angle, a rapid development of solitons is observed as the air velocity is increased preventing the waves to turn back. Finally, at high liquid Reynolds number, sudden and intermittent events are triggered consisting of very large amplitude waves that go back upwards very fast. These "slugs" either extend over the whole width of the channel or are very localized and can thus potentially evolve towards atomization. (C) 2017 Elsevier Ltd. All rights reserved