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.
We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a non-zero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set up parallel to the substrate surface - this nonlocal physical mechanism has a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid films that are hanging from the underside of the substrate; these films may drip depending on physical parameters, and we investigate whether a sufficiently strong electric field can suppres…
We consider the full 3D dynamics of a thin falling liquid film on a flat plate inclined at some non-zero angle to the horizontal. In addition to gravitational effects, the flow is driven by an electric field, which is normal to the substrate far from the flow. We study both the cases of overlying and hanging films, where the liquid rests above and below the substrate respectively. Starting with the Navier-Stokes equations coupled with electrostatics, a fully nonlinear 2D Benney equation for the interfacial dynamics is derived valid for waves that are long compared to the film thickness. The we…
We study the effect of insoluble surfactants on the wave dynamics of vertically-falling liquid films. We use three-dimensional numerical simulations and employ a hybrid interface-tracking/level-set method, taking into account Marangoni stresses induced by gradients of interfacial surfactant concentration. Our numerical predictions for the evolution of the surfactant-free, three-dimensional wave topology are validated against the experimental work of Park & Nosoko (2003). The addition of surfactants is found to influence significantly the development of horseshoe-shaped waves. At low Marangoni …
Instabilities arise in a number of flow configurations. One such manifestation is the development of interfacial waves in multiphase flows, such as those observed in the falling liquid film problem. Controlling the development of such instabilities is a problem of both academic and industrial interest. However, this has proven challenging in most cases due to the strong nonlinearity and high dimensionality of the underlying equations. In the present work, we successfully apply Deep Reinforcement Learning (DRL) for the control of the one-dimensional (1D) depth-integrated falling liquid film. In…
We present a control-volume approach for deriving a simplified model for the gravity-driven flow of an axisymmetric liquid film along a vertical fiber. The model accounts for gravitational, viscous, inertial and surface tension effects and results in a pair of coupled one-dimensional nonlinear partial differential equations for the film profile and average downward velocity as functions of time and axial distance along the fiber. Two versions of the model are obtained, one assuming a plug-flow velocity profile and a constant thin boundary layer thickness to model the drag force on the fluid, t…
The interaction that occurs between a light solid object and a horizontal soap film of a bamboo foam contained in a cylindrical tube is simulated in 3D. We vary the shape of the falling object from a sphere to a cube by changing a single shape parameter as well as varying the initial orientation and position of the object. We investigate in detail how the soap film deforms in all these cases, and determine the network and pressure forces that a foam exerts on a falling object, due to surface tension and bubble pressure respectively. We show that a cubic particle in a particular orientation exp…
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 solitar…
The numerical simulation of multiphase flows presents several challenges, namely the transport of different phases within de domain and the inclusion of capillary effects. Here, these are approached by enforcing a discrete physics-compatible solution. Extending our previous work on the discretization of surface tension [N. Valle, F. X. Trias, and J. Castro. An energy-preserving level set method for multiphase flows. J. Comput. Phys., 400:108991, 2020] with a consistent mass and momentum transfer a fully energy-preserving multiphase flow method is presented. This numerical technique is showcase…
Flow of a thin viscous film down a flat inclined plane becomes unstable to long wave interfacial fluctuations when the Reynolds number based on the mean film thickness becomes larger than a critical value (this value decreases as the angle of inclination with the horizontal increases, and in particular becomes zero when the plate is vertical). Control of these interfacial instabilities is relevant to a wide range of industrial applications including coating processes and heat or mass transfer systems. This study considers the effect of blowing and suction through the substrate in order to cons…
A conservation-consistent boundary condition for nonlinear models of soluble-surfactant-laden falling films
Centre for Applied Sciences (CAS), International Institute of Information Technology（国际信息技术学院应用科学中心）；Universite Claude Bernard Lyon 1, LMFA, UMR5509, CNRS, Ecole Centrale de Lyon, INSA Lyon（里昂第一大学，LMFA，UMR5509，CNRS，里昂中央理工学院，里昂国立应用科学学院）；Department of Mathematics, Vivekananda Mahavidyalaya（维韦卡南达学院数学系）
A thin liquid film falling on a uniformly heated horizontal plate spreads into fingering ripples that can display a complex dynamics ranging from continuous waves, nonlinear spatially localized periodic wave patterns (i.e. rivulet structures) to modulated nonlinear wavetrain structures. Some of these structures have been observed experimentally, however conditions under which they form are still not well understood. In this work we examine profiles of nonlinear wave patterns formed by a thin liquid film falling on a uniformly heated horizonal plate. In this purpose, the Benney model is conside…
We study the fluid mechanics of removing a passive tracer contained in small, viscous drops attached to a flat inclined substrate using thin gravity-driven film flows. A convective mass transfer establishes across the drop-film interface and the tracer in the drop diffuses into the film flow. The Peclet number for the tracer in the film is large. The Peclet number Pe_d in the drop varies from 0.01 to 1. The characteristic transport time in the drop is much larger than in the film. We model the mass transfer of the tracer from the drop bulk into the film using an empirical model based on Newton…
The flow of thin liquid films on inclined or vertical surfaces is one of immense importance, with applications spanning many types of process industries, due to the increased mass and heat transfer brought about by the presence of waves on film surfaces. While extensive research has been carried out in this area, many outstanding questions remain due to the complexity of the problem - a result of the extremely large aspect ratio and the three-dimensional nature. In this study, the evolution process of both naturally forming and forced waves on these thin liquid films is numerically studied usi…
A flow in which a thin film falls due to gravity on the inner surface of a vertical, rotating cylinder is investigated. This is performed using two-dimensional (2D) and three-dimensional (3D) direct numerical simulations, with a volume-of-fluid approach to treat the interface. The problem is parameterised by the Reynolds, Froude, Weber and Ekman numbers. The variation of the Ekman number ($Ek$), defined to be proportional of the rotational speed of the cylinder, has a strong effect on the flow characteristics. Simulations are conducted over a wide range of $Ek$ values ($0 \leq Ek \leq 484$) in…
The impact of droplets on an inclined falling liquid film is studied experimentally using high-speed imaging. The falling film is created on a flat substrate with controllable thicknesses and flow rates. Droplets with different sizes and speeds are used to study the impact process under various Ohnesorge and Weber numbers, and film Reynolds numbers. A number of phenomena associated with droplet impact are identified and analysed, such as bouncing, partial coalescence, total coalescence, and splashing. The effects of droplet size, speed, as well the film flow rate are studied culminating in the…
Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slope sufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating and heat transfer, which benefit from smooth and wavy interfaces, respectively. Here we discuss how the dynamics of the system are altered by feedback controls based on observations of the interface height, and supplied to the system via the perpendicular injection and suction of fluid through the wall. In this study, we model the system using both Benney and weighted…
We investigate nonlinear periodic and solitary two-dimensional rolling waves in a falling two-layer liquid film in the regime of non-zero Reynolds numbers. At any flow rate, a falling two-layer liquid film is known to be linearly unstable with respect to long-wave deformations of the liquid-air surface and liquid-liquid interface. Two different types of zero-amplitude neutrally stable waves propagate downstream without growing or shrinking: a zig-zag surface mode and a thinning varicose interface mode. Using a boundary-layer reduction of the Navier-Stokes equation, we investigate the onset, po…
The linear instability of a surfactant-laden two-layer falling film over an inclined slippery wall is analyzed under an influence of external shear which is imposed on the top surface of the flow. The free surface of the flow as well as the interface among the fluids are contaminated by insoluble surfactants. Dynamics of both the layers are governed by the Navier--Stokes equations, and the surfactant transport equation regulates the motion of the insoluble surfactants at the interface and free surface. Instability mechanisms are compared by imposing the external shear along and opposite to the…
Solutocapillary instability in slipping falling films
Centre for Applied Sciences (CAS), International Institute of Information Technology（国际信息技术学院应用科学中心）；Université Claude Bernard Lyon 1, LMFA, UMR 5509, CNRS, Ecole Centrale de Lyon, INSA Lyon（里昂第一大学）；Department of Mathematics, Vivekananda Mahavidyalaya（维韦卡南达学院数学系）
Accurate and computationally accessible models of liquid film flows allow for optimizing coating processes such as hot-dip galvanization and vertical slot-die coating. This paper extends the classic three-dimensional integral boundary layer (IBL) model for falling liquid films (FF) to account for a moving substrate (MS). We analyze the stability of the liquid films on vertically moving substrates in a linear and a nonlinear setting. In the linear analysis, we derive the dispersion relation and the temporal growth rates of an infinitesimal disturbance using normal modes and linearized governing…
The heat transfer characteristics of thin film flow over a hot sphere resulting from a cold vertical jet of liquid falling onto the surface has been investigated. The underlying physical features have been illustrated by numerical solutions of high accuracy based on the modified Keller box method. The solutions for film thickness distribution are good agreement with those obtained approximately by using the Pohlhausen integral momentum technique and observed experimentally by using water as working fluid, thus providing a basic confirmation of the validity of the results presented.
This paper is devoted to analytical solutions for the base flow and temporal stability of a liquid film driven by gravity over an inclined plane when the fluid rheology is given by the Carreau-Yasuda model, a general description that applies to different types of fluids. In order to obtain the base state and critical conditions for the onset of instabilities, two sets of asymptotic expansions are proposed, from which it is possible to find four new equations describing the reference flow and the phase speed and growth rate of instabilities. These results lead to an equation for the critical Re…
The jet wiping process is a cost-effective coating technique that uses impinging gas jets to control the thickness of a liquid layer dragged along a moving strip. This process is fundamental in various coating industries (mainly in hot-dip galvanizing) and is characterized by an unstable interaction between the gas jet and the liquid film that results in wavy final coating films. To understand the dynamics of the wave formation, we extend classic laminar boundary layer models for falling films to the jet wiping problem, including the self-similar integral boundary layer (IBL) and the weighted …
Solitary-like surface waves that originate from the spatio-temporal evolution of falling liquid films have been the subject of theoretical and experimental research due to their unique properties that are not readily observed in the physical system of other nature. Here we investigate, experimentally and theoretically, the dynamics of solitary-like surface waves in a liquid layer on an inclined plane that is subjected to a harmonic low-frequency vibration in the range from 30 to 50 Hz. We demonstrate that the vibration results in a decrease in the average and peak amplitude of the long solitar…
The ability to robustly and efficiently control the dynamics of nonlinear systems lies at the heart of many current technological challenges, ranging from drug delivery systems to ensuring flight safety. Most such scenarios are too complex to tackle directly and reduced-order modelling is used in order to create viable representations of the target systems. The simplified setting allows for the development of rigorous control theoretical approaches, but the propagation of their effects back up the hierarchy and into real-world systems remains a significant challenge. Using the canonical setup …
We consider the genesis and dynamics of interfacial instability in gas-liquid flows, using as a model the two-dimensional channel flow of a thin falling film sheared by counter-current gas. The methodology is linear stability theory (Orr-Sommerfeld analysis) together with direct numerical simulation of the two-phase flow in the case of nonlinear disturbances. We investigate the influence of three main flow parameters (density contrast between liquid and gas, film thickness, pressure drop applied to drive the gas stream) on the interfacial dynamics. Energy budget analyses based on the Orr-Somme…
