Gravity-driven Liquid Film Research Articles

Modelling falling film flow: an adjustable formulation

A new two-equation model for gravity-driven liquid film flow based on the long-wave expansion has been derived. The novelty of the model consists in using a base velocity profile combining parabolic (Ruyer-Quil & Manneville, Eur. Phys. J. B, vol. 15, issue 2, 2000, pp. 357–369) and ellipse (Usha et al., Phys. Fluids, vol. 32, issue 1, 2020, 013603) profile functions in the wall-normal coordinate. The dependence on a free parameter $A$ related to the eccentricity of an ellipse serves as an adjustable parameter. The resulting models are consistent at $O(\varepsilon )$ for inertia terms and at $O(\varepsilon ^2)$ for viscous diffusion effects, and predict accurately the primary instability. Appropriate tuning of the adjustable parameter helps to recover accurate predictions for the asymptotic wave celerity of nonlinear solitary waves. Further, the model is shown to capture the closed separation vortices that can form underneath the troughs of precursory capillary ripples.

Thermal analysis for a gravity-driven liquid film along an inclined porous substrate

This paper investigates the entropy generation in a thin film fluid flow along a heated inclined substrate. The free end of the thin film is assumed to be exposed to adiabatic condition. The mathematical formulation for the fluid and heat flow leads to a dimensionless Boundary Value Problem (BVP) together with a partial differential equation (PDE). The exact solution of the BVP problem was obtained and the analytical solution for the PDE was obtained with the use of separation of variables. These solutions were used to profile the entropy generation in the thin film flow. The result earlier obtained in the absence of porous permeability was validated as a special case of the work and our contributions to knowledge were highlighted in the porous substrate case.

Stability of gravity-driven liquid films overflowing microstructures with sharp corners

We are concerned with the stability of thin liquid films overflowing single microstructures with sharp corners. The microstructures were of rectangular and triangular shape. Their heights and widths were 0.25, 0.5 and 0.75 times the Nusselt film thickness. To observe smooth, wavy and very unstable films we performed simulations with Reynolds numbers ranging from 10 to 70. The dynamics of the liquid film and the overflowing gas phase were described by the coupling between the Cahn-Hilliard and Navier-Stokes equations. The resulting model forms a very tightly coupled and nonlinear system of equations. Therefore we carefully selected the solution strategy to enable efficient and accurate large-scale simulations. Our results showed that the formation of waves was shifted to higher Reynolds numbers compared to the film on a smooth surface. If waves were finally formed the microstructures led to irregular waves. Our results indicate a great influence of the microstructure's shape and dimension on the stability of the overflowing liquid film.

Stability of liquid film flow laden with the soluble surfactant sodium dodecyl sulphate: predictions versus experimental data

Gravity-driven liquid film flows laden with a soluble surfactant are considered, and aqueous solutions of sodium dodecyl sulphate (SDS) are taken as a case-study. Literature measurements of the critical Reynolds number for the onset of instability are set in perspective with predictions of linear stability theory. The theory is based on a Frumkin model of adsorption equilibrium, modified by the inclusion of finite compressibility of the adsorbed monolayer. Quantitative comparison between data and theory is first attempted in the limit of infinite wavelength. Though wave characteristics are satisfactorily predicted, the theoretical critical Reynolds number is an order of magnitude below measurements. This discrepancy is understood in terms of the large difference between momentum and mass diffusivities and indicates that the assumption of infinite wavelength is far more restrictive for the mass transfer than for the flow problem. Finite-wavelength effects are taken into account by numerical solution of the Orr–Sommerfeld eigenvalue problem, leading to predictions of maximum stabilization in good agreement with the measurements. Introduction of realistic values of monolayer compressibility improves further the agreement at high surfactant loadings. Finally, a strong stabilizing effect of salinity is confirmed.

Mixed convection in gravity-driven thin nano-liquid film flow with homogeneous–heterogeneous reactions

A modified nanofluid model for homogeneous–heterogeneous reactions in a gravity-driven liquid film is proposed based on the assumptions of the homogeneous reaction governed by isothermal cubic autocatalytic kinetics and the heterogeneous reaction given by first-order kinetics. The Buongiorno model is introduced to describe behaviors of nanofluids with the Boussinesq approximation being used for simplification of the buoyancy term. Multiple solutions are captured, whose stabilities are checked and discussed based on the theory of Sturm–Liouville. The influence of various physical parameters on important physical quantities is presented for different cases, including buoyancy assisting, no buoyancy, and buoyancy opposing. Different from previous studies on chemical reactions in which reactants are presumed in nanometer scale, we assume that the chemical species are of regular size, which react with each other in a nanofluid. This configuration makes our model physically more realistic than previous ones.

Fingering Instability of a Gravity-Driven Thin Film Flowing Down a Vertical Tube with Wall Slippage

Inspired by the antiwetting property of pitcher plants, specialists have designed different functional material with slippery surfaces, and a directional slippery surface has been fabricated. This paper considers a gravity-driven liquid film coating the interior surface of a vertical tube, and different slippery lengths in the azimuthal direction and the axial direction are taken into account. The evolution equation of coating flow is derived using the thin film model, and time responses for two dimensional flow are calculated. Linear stability analysis (LSA) is given based on the traveling wave solutions, demonstrating that the axial slippery effect suppresses the flow instability and causes a larger traveling wave speed. Simultaneously, the azimuthal slippery effect plays a destabilizing role for perturbations with small wavenumbers and it is stabilizing for large wavenumbers. Direct simulations of the fingering flow patterns agree well with the linear stability analysis. Our results offer insight into the influence of wall slippage on the flow stability of liquid in petroleum engineering.

Pancake making and surface coating: Optimal control of a gravity-driven liquid film

This paper investigates the flow of a solidifying liquid film on a solid surface subject to a complex kinematics, a process relevant to pancake making and surface coating. The flow is modeled using the lubrication approximation, with a gravity force whose magnitude and direction depend on the time-dependent orientation of the surface. Solidification is modeled with a temperature-dependent viscosity. Because the flow eventually ceases as the liquid film becomes very viscous, the key question this study aims to address is: what is the optimal surface kinematics for spreading the liquid layer uniformly? Two methods are proposed to tackle this problem. In the first one, the surface kinematics is assumed a priori to be harmonic and parameterized. The optimal parameters are inferred using the Monte-Carlo method. This ``brute-force'' approach leads to a moderate improvement of the film uniformity compared to the reference case when no motion is imposed to the surface. The second method is formulated as an optimal control problem, constrained by the governing partial differential equation, and solved with an adjoint equation. Key benefits of this method are that no assumption is made on the form of the control, and that significant improvement in thickness uniformity are achieved with a comparatively smaller number of evaluations of the objective function.

Modelling gravity-driven film flow on inclined corrugated substrate using a high fidelity weighted residual integral boundary-layer method

A computational investigation is conducted concerning the stability of free-surface gravity-driven liquid film flow over periodic corrugated substrate. The underpinning mathematical formulation constitutes an extension of the weighted residual integral boundary-layer (WIBL) method proposed by Ruyer-Quil and Manneville [“Improved modeling of flows down inclined planes,” Eur. Phys. J. B 15(2), 357–369 (2000)] and D’Alessio et al. [“Instability in gravity-driven flow over uneven surfaces,” Phys. Fluids 21(6), 062105 (2009)] to include third- and fourth-order terms in the long-wavelength expansion. Steady-state solutions for the free-surface and corresponding curves of neutral disturbances are obtained using Floquet theory and validated against corresponding experimental data and full Navier-Stokes (N-S) solutions. Sinusoidal and smoothed rectangular corrugations with variable steepness are considered. It is shown that the model is capable of predicting characteristic patterns of stability, including short-wave nose and isles of stability/instability as reported experimentally for viscous film flow over inclined topography, providing an attractive trade-off between the accuracy of a full N-S computation and the efficiency of an integral method. The range of parameter values for which the WIBL model remains valid is established; in particular, it is shown that its accuracy decreases with the Reynolds number and corrugation amplitude, but increases with the steepness parameter and ratio of wavelength to capillary length.

Experimental Investigation of Local Film Thickness and Velocity Distribution Inside Falling Liquid Films on Corrugated Structured Packings

Understanding the complex fluid dynamics of the liquid phase in structured packing is crucial for the prediction of the separation performance of packed tower and for further optimization of the packing geometry. Detailed numerical investigations are useful to study a wide range of parameters, but require high computational effort and validation is required. Therefore, in this study, experimental investigations on the gravity driven liquid film flow phenomena over corrugated structured packing geometries are presented and evaluated. Due to the fragile nature of thin liquid films, nonintrusive optical measurements where performed through the backside of transparent moldings of packing geometries, thereby reducing optical distortion. To enable the investigation of three-dimensional flow phenomena a µ-Stereoscopic Particle Image Velocimetry (µSPIV) was adapted for thin liquid films. Measurements are performed on MONTZ-Pak B1.250 macro-structure with a smooth surface or a micro-structured surface at different locations. Values of film thicknesses and velocity distributions inside the falling liquid film are discussed in detail.

Modelling of an innovative liquid rotational moulding process

An innovative liquid rotational moulding process aiming at using standard injection-moulding polymer grades, shortening cycle times and at accessing more higher-end markets is studied. In this new process, the polymer is melted beforehand in a conventional injection unit and injected into the mould rotating around two axes. This work focus on modelling the liquid polymer flow occurring in this process, as well as providing guidance on the choice of process parameters. First, a stability criterion for rimming flow is derived, defining the process and material parameters space available for a functional process. From this criterion, it appears that initial thickness, and density have a destabilising effect, while liquid viscosity and angular velocity have a stabilising effect. Next, using Lax-Friedrichs scheme, the one-dimensional transient gravity driven liquid film is numerically modelled, linking the total extent of spreading to the frequency of orientation reversals and to the major and minor angular velocities ratio : the closer this ratio to one, the larger the extent of spreading. Then, Computational Fluid Dynamics (CFD) simulations are carried out to take into account three-dimensional features in the moulding of a cube part. Finally, comparison with trials performed on the moulding of the same cube part validates entirely this approach. In particular, some peculiar features of spreading can only be explained and reproduced when employing three-dimensional CFD simulations.

Critical heat flux in locally heated liquid film moving under the action of gas flow in a mini-channel

Thin and ultra thin liquid films driven by a forced gas/vapor flow (stratified or annular flows), i.e. shear-driven liquid films in a narrow channel, is one of the promising candidate for the thermal management of advanced semiconductor devices with high local heat release. In experiments performed in this paper with locally heated shear-driven liquid films of water the effect of various conditions, such as flow rates of liquid and gas and channel height, on critical heat flux (CHF) was investigated. In experiments the record value of CHF as high as 540 W/cm2 has been achieved. The heat spreading into the substrate and the heat loses into the atmosphere in total don't exceed 30% at heat fluxes higher than 200 W/cm2. Comparison of shear-driven liquid films and gravity-driven liquid films showed that CHF in shear-driven films up to 10 times higher than in gravity-driven liquid films. Thus, prospect of using shear- driven films of water in modern cooling systems of semiconductor devices was confirmed.

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

We propose consistent scaling of solitary waves on inertia-dominated falling liquid films, which accurately accounts for the driving physical mechanisms and leads to a self-similar characterization of solitary waves. Direct numerical simulations of the entire two-phase system are conducted using a state-of-the-art finite volume framework for interfacial flows in an open domain that was previously validated against experimental film-flow data with excellent agreement. We present a detailed analysis of the wave shape and the dispersion of solitary waves on 34 different water films with Reynolds numbers Re=20-120 and surface tension coefficients σ=0.0512-0.072 N m(-1) on substrates with inclination angles β=19°-90°. Following a detailed analysis of these cases we formulate a consistent characterization of the shape and dispersion of solitary waves, based on a newly proposed scaling derived from the Nusselt flat film solution, that unveils a self-similarity as well as the driving mechanism of solitary waves on gravity-driven liquid films. Our results demonstrate that the shape of solitary waves, i.e., height and asymmetry of the wave, is predominantly influenced by the balance of inertia and surface tension. Furthermore, we find that the dispersion of solitary waves on the inertia-dominated falling liquid films considered in this study is governed by nonlinear effects and only driven by inertia, with surface tension and gravity having a negligible influence.

Investigation of heat transfer enhancement inside developing gravity-driven films along a vertical permeable plate subject to nonuniform suction

Flow and heat transfer through the boundary layers of a falling liquid film on a vertical permeable plate subject to nonuniform suction flow are analyzed in this work. The continuity, momentum, and energy equations are transformed to nonsimilar equations and solved using a validated implicit and iterative finite difference method. Increases in the Froude number, Galilei number, and the dimensionless average suction velocity are found to increase the skin friction coefficient, the Nusselt number, and the heat transfer enhancement ratios. These enhancement ratios are noticed to increase at the plate exit as the suction velocity power-law index increases. The Froude number for uniform suction case required to attain the same enhancement ratios due to nonuniform distribution of this suction flow is found to increase as the Galilei number, average suction velocity, and power-law index increase. It is found that the upper most values of the enhancement factors in heat transfer and Froude number are [Formula: see text] folds when the suction power-law index is equal to [Formula: see text]. This work demonstrated that significant heat transfer enhancement inside developing gravity-driven liquid films is attainable when properly distributing the suction velocity along the plate.

Understanding of the liquid overflow behavior inside micro-structured falling film reactors based on a wetting approach

This study investigates the behavior of gravity-driven liquid films inside falling film micro-reactors and focuses on the conditions necessary to obtain and break-up the complete liquid overflow from all of the channels. Micro-structured falling film reactors containing 6–10 straight parallel rectangular micro-channels were designed, constructed and tested. The main objective was to show experimentally the influence of liquid wettability on the hydrodynamic behavior. To achieve this, we explored the fluidic motion as a function of several parameters, including micro-channel size, the structural materials of the reactor (stainless steel, fluorocarbon and silicon oxide), liquid nature (water, water–ethanol mixtures and aqueous solutions of a ionic sodium dodecyl sulfate surfactant) and concentration levels. The major parameters that control the hydrodynamic behavior are the liquid/solid advancing and receding contact angles. The advancing contact angle governs the filling and the overflow of the channels, while the receding contact angle drives the characteristics of the break-up of the total overflow. Through experimentation, we identified a relationship between the contact angles and the resulting liquid flow patterns. We observed that spreading increases with higher ethanol content levels, surfactant concentration or by using a silicon oxide reactor. The flow rate required to achieve or break-up the total overflow decreased as a result. Finally, all of these results can be summarized by two master curves relating the critical flow rates to achieve and break-up the overflow with the advancing and receding contact angles, respectively. The channel dimensions also play a role in the hydrodynamic behavior, since the reduction of the channel width and distance between successive channels leads to a decrease in the critical flow rates.

Second law analysis of a gravity-driven liquid film flowing along an inclined plate subjected to constant wall temperature

The second law analysis of a gravity-driven liquid having temperature-dependent viscosity and flowing along an inclined heated plate was studied. The computational fluid dynamics (CFD) package FLUENT was used to obtain the velocity and temperature gradients, and the results were used to determine the entropy generation rate. Different cases of inclination angle and dimensionless parameters were investigated. There is significant difference in the entropy generation rate magnitude in comparison to the entropy generation for constant physical properties case. The entropy generation rate has its maximum value near the plate surface and decreases sharply towards

Wavy liquid films in interaction with a confined laminar gas flow

AbstractA low-dimensional model capturing the fully coupled dynamics of a wavy liquid film in interaction with a strongly confined laminar gas flow is introduced. It is based on the weighted residual integral boundary layer approach of Ruyer-Quil & Manneville (Eur. Phys. J.B, vol. 15, 2000, pp. 357–369) and accounts for viscous diffusion up to second order in the film parameter. The model is applied to study two scenarios: a horizontal pressure-driven water film/air flow and a gravity-driven liquid film interacting with a co- or counter-current air flow. In the horizontal case, interfacial viscous drag is weak and interfacial waves are primarily driven by pressure variations induced by the velocity difference between the two layers. This produces an extremely thin interfacial shear layer which is pinched at the main and capillary wave humps, creating several elongated vortices in the wave-fixed reference frame. In the capillary wave region preceding a large wave hump, flow separation occurs in the liquid in the form of a vortex transcending the liquid–gas interface. For the gravity-driven film, a twin vortex (in the wave-fixed reference frame) linked to the occurrence of rolling waves has been identified. It consists of the known liquid-side vortex within the wave hump and a previously unknown counter-rotating gas-side vortex, which are connected by the same interfacial stagnation points. At large counter-current gas velocities, interfacial waves on the falling liquid film are amplified and cause flooding of the channel in a noise-driven scenario, while this can be delayed by forcing regular waves at the most amplified linear wave frequency. Our model is shown to exactly capture the long-wave linear stability threshold for the general case of two-phase channel flow. Further, for the two considered scenarios, it predicts growth rates and celerity of linear waves in convincing agreement with Orr–Sommerfeld calculations. Finally, model calculations of nonlinear interfacial waves are in good agreement with film thickness and velocity measurements as well as streamline patterns in both phases obtained from direct numerical simulations.

Evaporation and flow dynamics of thin, shear-driven liquid films in microgap channels

Thin and ultra-thin shear-driven liquid films in a narrow channel are a promising candidate for the thermal management of advanced semiconductor devices in earth and space applications. Such flows experience complex, and as yet poorly understood, two-phase flow phenomena requiring significant advances in fundamental research before they could be broadly applied. This paper focuses on the results obtained in experiments with locally heated shear-driven liquid films in a flat mini-channel. A detailed map of the flow sub-regimes in a shear-driven liquid film flow of water and FC-72 have been obtained for a 2 mm channel operating at room temperature. While the water film can be smooth under certain liquid/gas flow rates, the surface of an intensively evaporating film of FC-72 is always distorted by a pattern of waves and structures. It was found, that when heated the shear-driven liquid films are less likely to rupture than gravity-driven liquid films. For shear-driven water films the critical heat flux was found of up to 10 times higher than that for a falling film, which makes shear-driven films (annular or stratified two-phase flows) more suitable for cooling applications than falling liquid films.

Development of a PLIC-VOF method for the dynamic simulation of entry region flow in a laminar falling film

The present work describes a numerical procedure to simulate the development of hydrodynamic entry region in a gravity-driven laminar liquid film flow over an inclined plane. It provides a better insight into the physics of developing film in entry region. A novel numerical approach is proposed which has the potential to provide solutions for the complex physics of liquid film spreading on solid walls. The method employs an incompressible flow algorithm to solve the governing equations, a PLIC-VOF method to capture the free surface evolution and a continuum surface force (CSF) model to include the effect of surface tension. To account for the moving contact line on the solid substrate, a precursor film model based wall treatment is implemented. Liquid film flow has been simulated for the Reynolds number range of 5 ≤ Re ≤ 37.5, and the predicted results are found to agree well with the available analytical and experimental data.

Second-law analysis of laminar nonnewtonian gravity-driven liquid film along an inclined heated plate with viscous dissipation effect

A second-law analysis of a gravity-driven film of non-Newtonian fluid along an inclined heated plate is investigated. The flow is assumed to be steady, laminar and fully-developed. The upper surface of the liquid film is considered to be free and adiabatic. The effect of heat generation by viscous dissipation is included. Velocity, temperature and entropy generation profiles are presented. The effects of the flow behaviour index, the Brinkman number and the group parameter on velocity, temperature and entropy generation number are discussed. The results show that velocity profile depends largely on the flow behaviour index. They are flat near the free surface for pseudoplastic fluids and linear for dilatant fluids. Temperature profiles are higher for higher flow behaviour index and Brinkman number. The entropy generation number increases with Brinkman number and the group parameter because of the heat generated by the viscous dissipation effect. For pseudoplastic fluids, the irreversibility is dominated by heat transfer, whereas, for dilatant fluids, irreversibility due to fluid friction is more dominant.

Mathematical modeling of moving contact lines in heat transfer applications

We provide an overview of research on the mathematical modeling of apparent contact lines in non-isothermal systems conducted over the past several decades and report a number of recent developments in the field. The latter involve developing mathematical models of evaporating liquid droplets that account not only for liquid flow and evaporation, but also for unsteady heat conduction in the substrate. The droplet is placed on a flat heated solid substrate and is assumed to be in contact with a saturated vapor. Furthermore, we discuss a careful comparison between mathematical models and experimental work that involves simultaneous measurement of shapes of evaporating droplets and temperature profiles in the solid substrate. The latter is accomplished using thermochromic liquid crystals. Applications to new research areas, such as studies of the effect of evaporation on fingering instabilities in gravity-driven liquid films, are also discussed.