The nonlinear stability of falling film flow down an inclined flat plane is investigated when an electric field acts normal to the plane. A systematic asymptotic expansion is used to derive a fully nonlinear long-wave model equation for the scaled interface, where higher-order terms must be retained to make the long-wave approximation valid for long times. The effect of the electric field is to introduce a non-local term which comes from the potential region above the liquid film. This term is always linearly destabilizing and produces growth rates proportional to the cubic power of the wavenumber – surface tension is included and provides a short wavelength cutoff. Even in the absence of an electric field, the fully nonlinear equation can produce singular solutions after a finite time. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto–Sivashinsky equation. This equation has solutions which exist for all time and allows for a complete study of the nonlinear behaviour of competing physical mechanisms: long-wave instability above a critical Reynolds number, short-wave damping due to surface tension and intermediate growth due to the electric field. Through a combination of analysis and extensive numerical experiments, we find parameter ranges that support non-uniform travelling waves, time-periodic travelling waves and complex nonlinear dynamics including chaotic interfacial oscillations. It is established that a sufficiently high electric field will drive the system to chaotic oscillations, even when the Reynolds number is smaller than the critical value below which the non-electrified problem is linearly stable. A particular case of this is Stokes flow.

We investigate the dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film. The solution of the full system of equations describing the gas-liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to pressure and tangential stresses at the interface imposed by the turbulent gas in closed form. We then analyse the liquid film flow under the influence of these perturbations and derive a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the KuramotoSivashinsky equation with an additional term due to the presence of the turbulent gas. This additional term is dispersive and destabilising (for the counter-current case; stabilizing in the co-current case). We also combine the long-wave approximation with a weighted-residual technique to obtain an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly. We confirm the existence of large-amplitude stationary waves by computing periodic travelling waves for the integral-boundary-layer approximation and we corroborate our travelling-wave results by time-dependent computations.

Absolute and convective instabilities in counter-current gas-liquid film flows. Mechanics, 763, Additional Information: Journal of Fluid• This is the accepted manuscript of an article subsequently published in We consider a thin liquid film flowing down an inclined plate in the presence of a countercurrent turbulent gas. By making appropriate assumptions, Tseluiko & Kalliadasis (2011) developed low-dimensional non-local models for the liquid problem, namely a long-wave model and a weighted integral-boundary-layer model, that incorporate the effect of the turbulent gas. By utilising these models, along with the Orr-Sommerfeld problem formulated using the full governing equations for the liquid phase and associated boundary conditions, we explore the linear stability of the gas-liquid system. In addition, we devise a generalised methodology to investigate absolute and convective instabilities in the nonlocal equations describing the gas-liquid flow. We observe that at low gas flow rates, the system is convectively unstable with the localised disturbances being convected downwards. As the gas flow rate is increased, the instability becomes absolute and localised disturbances spread across the whole domain. As the gas flow rate is further increased, the system again becomes convectively unstable with the localised disturbances propagating upwards. We find that the upper limit of the absolute instability region is close to the 'flooding' point associated with the appearance of large-amplitude standing waves, as obtained in Tseluiko & Kalliadasis (2011), and our analysis can therefore be used to predict the onset of flooding. We also find that an increase in the angle of inclination of the channel requires an increased gas flow rate for the onset of absolute instability. We generally find good agreement between the results obtained using the full equations and the reduced models. Moreover, we find that the weighted integral-boundary-layer model generally provides better agreement with the results for the full equations than the longwave model. Such an analysis is important for understanding the ranges of validity of the reduced model equations. In addition, a comparison of our theoretical predictions with the experiments of Zapke & Kröger (2000a) shows a fairly good agreement. We supplement our stability analysis with time-dependent computations of the linearised weighted integral-boundary-layer model. To provide some insight into the mechanisms of instability, we perform an energy budget analysis.

When a plate is withdrawn from a liquid bath a coating layer is deposited whose thickness and homogeneity depend on the velocity and the wetting properties of the plate. Using a long-wave mesoscopic hydrodynamic description that incorporates wettability via a Derjaguin (disjoining) pressure we identify four qualitatively different dynamic transitions between microscopic and macroscopic coatings that are out-of-equilibrium equivalents of well known equilibrium unbinding transitions. Namely, these are continuous and discontinuous dynamic emptying transitions and discontinuous and continuous dynamic wetting transitions. We uncover several features that have no equivalent at equilibrium.The equilibrium and non-equilibrium behaviour of mesoscopic and macroscopic drops, meniscii and films of liquid in contact with static or moving solid substrates is not only of fundamental interest but also crucial for a large number of modern technologies. On the one hand, the equilibrium behaviour of films, drops and meniscii is studied by means of statistical physics. A rich substrate-induced phase transition behaviour is described even for simple liquids, e.g., related to wetting and emptying transitions that both represent unbinding transitions. In the former case the thickness of an adsorption layer of liquid diverges continuously or discontinuously at a critical temperature or strength of substrate-liquid interaction, i.e., the liquid-gas interface of the film unbinds from the liquid-solid interface [1]. In the case of the emptying transition a macroscopic meniscus in a tilted slit capillary develops a tongue (or foot) along the lower wall of a length that diverges logarithmically at a critical slit width, i.e., the tip of the foot unbinds from the meniscus [2].

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.

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.

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabalized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.

We study a nonlinear nonlocal evolution equation describing the hydrodynamics of thin films in the presence of normal electric fields. The liquid film is assumed to be perfectly conducting and to completely wet the upper or lower surface of a horizontal flat plate. The flat plate is held at constant voltage, and a vertical electric field is generated by a second parallel electrode kept at a different constant voltage and placed at a large vertical distance from the bottom plate. The fluid is viscous, and gravity and surface tension act. The equation is derived using lubrication theory and contains an additional nonlinear nonlocal term representing the electric field. The electric field is linearly destabilizing and is particularly important in producing nontrivial dynamics in the case when the film rests on the upper side of the plate. We give rigorous results on the global boundedness of positive periodic smooth solutions, using an appropriate energy functional. We also implement a fully implicit numerical scheme and perform extensive numerical experiments. Through a combination of analysis and numerical experiments we present evidence for the global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena, which are also observed in hanging films when electric fields are absent.