Stability of evaporating two-layered liquid film in the presence of surfactant—III. Non-linear stability analysis (original) (raw)
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Chemical Engineering Science, 1998
In this study we consider the stability of horizontal two-layered liquid film attached to a heated solid substrate. The film can contain surfactant that is soluble in both liquid phases. The evaporation of solvent from the upper film is also taken into account. Thus, the two-layered film can exhibit both thermocapillary and Marangoni instabilities coupled with the effect of solvent mass loss. The problem is solved in the framework of the lubrication approximation. We derive a system of partial differential equations describing the evolution of long-wave disturbances in the presence of surfactant and evaporation. Appropriate rescaling is proposed and a numerical analysis of the dimensionless groups for particular system of water-light oil film upon a horizontal PVC plate is performed. The model allows one to investigate the role of different factors on the film stability: the surfactant concentration and distribution coefficient, the critical concentrations of micellization, the surface viscosities, the adsorption isotherms of the surfactant at the liquid surfaces and the intensity of evaporation. Based on this model the full linear analysis of the stability is given in Part II [see Danov et al. (1997b) Chem. Engng Sci., submitted]. The non-linear effects are also taken into account in Part III [see Chem. Engng Sci. submitted], where these effects are studied numerically for a particular case of PVC/tetrachlorethane/water/vapour system.
Stability of evaporating two-layered liquid film in the presence of surfactant—II. Linear analysis
Chemical Engineering Science, 1998
In this study we consider the stability of horizontal two-layered liquid film attached to a heated solid substrate. The film can contain surfactant that is soluble in both liquid phases. The evaporation of solvent from the upper film is also taken into account. Thus, the two-layered film can exhibit both thermocapillary and Marangoni instabilities coupled with the effect of solvent mass loss. The problem is solved in the framework of the lubrication approximation. We derive a system of partial differential equations describing the evolution of long-wave disturbances in the presence of surfactant and evaporation. Appropriate rescaling is proposed and a numerical analysis of the dimensionless groups for particular system of water-light oil film upon a horizontal PVC plate is performed. The model allows one to investigate the role of different factors on the film stability: the surfactant concentration and distribution coefficient, the critical concentrations of micellization, the surface viscosities, the adsorption isotherms of the surfactant at the liquid surfaces and the intensity of evaporation. Based on this model the full linear analysis of the stability is given in Part II [see Danov et al. (1997b) Chem. Engng Sci., submitted]. The non-linear effects are also taken into account in Part III [see Chem. Engng Sci. submitted], where these effects are studied numerically for a particular case of PVC/tetrachlorethane/water/vapour system.
Physics of Fluids, 1998
The dynamics of an evaporating wetting liquid film in the presence of dissolved surfactant is investigated. The solid substrate is planar and is subjected to heating. The liquid-vapor interface is a two-dimensional continuum characterized by specific adsorption, interfacial viscosity, and surface tension, which depend on the surfactant subsurface concentration and temperature. In the case of small density, viscosity, and thermal conductivity of the vapor phase ͑compared to the respective values for the liquid phase͒, at small Reynolds and large Peclet numbers and for thin films, the lubrication approximation model can be applied. The effect of the van der Waals disjoining pressure is taken into account. The appearing dimensionless groups, defined in terms of the real physical parameters, can vary by several orders of magnitude depending on the film initial thickness, temperature difference, and type of surfactants. The developed linear theory describes the competition among the various instabilities. The numerical solution of the evolution equation provides information about the critical film thickness, critical lateral wave number, and time for rupture. The influence of the interfacial mass loss due to evaporation, the van der Waals attraction, the Marangoni effects due to thermal and concentration gradients, and the interfacial viscous friction upon the critical film thickness is discussed.
1998
human eye, in which the conditions for film rupture represent We consider an evaporating liquid film which lies on a planar the main objective of investigation. Numerous studies have heated solid substrate. The film contains a dissolved surfactant at been devoted to this problem, either in the geometry of free a high concentration, so that micellar aggregates exist in the bulk. films or for liquid layers on solid substrates. Scheludko (1) Linear stability analysis of this system is performed by investigatsuggested the idea that rupture instability results from ampliing the time evolution of the amplitude of fluctuation waves. The fication of spontaneous fluctuations in the shape of the fluid liquid-vapor interface is regarded as a two-dimensional continuum
STABILITY ANALYSIS OF EVAPORATING THIN LIQUID FILMS IN THE PRESENCE OF SURFACTANT
This research is dedicated for analyzing the stability of thin liquid film in the presence of heat and surfactant, the fluid flow of thin liquid films represents by Navier-Stokes equation and equation of continuity in two dimensional forms as shown in figure (1). We find that the stability of films occur when the Prandtl and Schmidt numbers are greater than zero,otherwise the film become unstable.
A mechanism of Marangoni instability in evaporating thin liquid films due to soluble surfactant
Physics of Fluids, 2010
In film coating and other applications involving thin liquid films, surfactants are typically employed to suppress the usually undesirable instabilities driven by surface phenomena. Yet, in the present study a mechanism of Marangoni instability in evaporating thin films is presented and analyzed, which has its origin on the effects of a soluble surfactant. As the film thins due to evaporation, thickness perturbations lead to surfactant concentration perturbations, which in turn drive film motion and tend to enhance uneven drying. A thin-film analysis is applied and evolution equations for the film thickness and the surfactant concentration are derived and analyzed by the techniques of linear stability and numerical simulation. In the linear analysis a nonautonomous system is obtained for the film thickness and surfactant concentration perturbations, which shows that the instability will manifest itself provided that an appropriate Marangoni number is relatively large and the surfactant solubility in the bulk is large as well. On the other hand, low solubility in the bulk, diffusion, and the effect of surfactant on interfacial mobility through the surface viscosity are found to suppress disturbance growth. Direct numerical simulations of the full nonlinear evolution equations confirm those results and add to the picture obtained for the physical system behavior. Estimates of the relevant dimensionless parameters suggest that the conditions for instability may be met in relatively thick films, on the order of tens of microns, for which the effects of molecular forces and disjoining pressure are not dominant. Moreover, the stabilizing effects of diffusion and interfacial mobility are not likely to become significant unless the films are much thinner, i.e., on the order of 1 m or below.
Dynamics of a horizontal thin liquid film in the presence of reactive surfactants
Physics of Fluids, 2007
We investigate the interplay between a stable horizontal thin liquid film on a solid substrate and an excitable or bistable reactive mixture on its free surface. Their coupling is twofold. On the one hand, flow in the film transports the reacting surfactants convectively. On the other hand, gradients in the surfactant concentration exert Marangoni stresses on the free surface of the film. A reduced model is derived based on the long-wave approximation. We analyze the linear stability of the coupled system as well as the nonlinear behavior, including the propagation of solitary waves, fronts, and pulses. We show, for instance, that the coupling of thin film hydrodynamics and surfactant chemistry can either stabilize instabilities occurring in the pure chemical system, or in a regime where the pure hydrodynamic and chemical subsystems are both stable, the coupling can induce instabilities.
Stability of draining plane-parallel films containing surfactants
Advances in Colloid and Interface Science, 2002
The stability of partially mobile draining thin liquid films with respect to axisymmetric Ž fluctuations was studied. The material properties of the interfaces Gibbs elasticity, surface. and bulk diffusions were taken into account. When studying the long wave stability of films, the coupling between the drainage and perturbation flows was considered and the lubrication approximation was applied. Two types of wave modes were examined: radially-bounded and unbounded waves. The difference between the thickness of loss of stability, h , the st transitional thickness, h , at which the critical wave causing rupture becomes unstable, and tr the critical thickness, h , when the film ruptures, is demonstrated. Both the linear and the cr non-linear theories give h) h) h. The numerical results show that the interfacial st tr cr mobility does not significantly influence the thickness of the draining film rupture. The interfacial tension and the disjoining pressure are the major factors controlling the critical thickness. The available experimental data for critical thicknesses of foam and emulsion films show excellent agreement with the theoretical predictions. The important role of the electromagnetic retardation term in the van der Waals interaction is demonstrated. Other published theories of the film stability are discussed.
Instabilities in evaporating liquid layer with insoluble surfactant
Physics of Fluids, 2013
The stability of an evaporating liquid layer with insoluble surfactant distributed over the free deformable surface is studied theoretically. The insoluble surfactant hinders the evaporation, and mass flux through the interface is a decreasing function of surfactant concentration. Density, viscosity, and thermal conductivity of the gaseous phase are assumed to be small compared with those of the liquid phase, and a onesided model is applied. A system of nonlinear equations is obtained using the longwave approximation and the assumption of a slow time evolution. These equations incorporate basic physical effects which take place in the system. Linear stability analysis of the base state is performed for long-wave disturbances in the framework of the frozen interface approximation. The cases of quasi-equilibrium evaporation (when the interfacial temperature equals the equilibrium one) and nonequilibrium evaporation are considered. In addition to a monotonic instability mode, an oscillatory mode has been found.
Stability of surfactant-laden liquid film flow over a cylindrical rod
Physical Review E, 2020
The stability of surfactant-laden liquid film flow over a cylindrical rod is examined in creeping flow limit using standard temporal linear stability analysis. The clean film flow configuration (i.e., in absence of surfactant) is well-known to become unstable owing to Rayleigh-Plateau instability of cylindrical liquid interfaces. Previous studies demonstrated that for a static liquid film (i.e., zero basic flow) coating a rod, the presence of interfacial surfactant decrease the growth of Rayleigh-Plateau instability, but is unable to suppress it completely. Further, the presence of interfacial surfactant is known to introduce an additional mode, referred to as surfactant mode in the present work. To the best of our knowledge, the stability of surfactant mode has not been analyzed in the context of cylindrical film flows. Thus, we reexamined the stability of surfactant-laden cylindrical liquid film flow to analyze the stability behavior of the above said two modes when the basic flow is turned on. The present study reveals that the incorporation of basic flow in stability analysis leads to the complete suppression of Rayleigh-Plateau instability due to the presence of interfacial surfactants as compared to the partial suppression obtained for a stationary liquid film. Three nondimensional parameters appear for this problem: Bond number (denoted as Bo) which characterizes the strength of basic flow, Marangoni number (denoted as Ma) which signifies the presence of surfactant, and ratio of rod radius to film thickness denoted as S. In creeping flow limit, the characteristic equation is quadratic with one root belonging to Rayleigh-Plateau mode and the other to surfactant mode. We first carried out an asymptotic analysis to independently capture the eigenvalues corresponding to both the modes in limit of long-wave disturbances. The long-wave results show that the Rayleigh-Plateau instability is completely suppressed on increasing the Marangoni number above a critical value while the surfactant mode always remains stable in low wave-number limit. The continuation of long-wave results to arbitrary wavelength disturbances show that the suppression of Rayleigh-Plateau instability mode still holds, however, the surfactant mode becomes unstable at sufficiently high values of Marangoni number. Further, this surfactant mode instability shifts toward low wave numbers with critical Marangoni number for instability scaling with wave number in a particular fashion. We used this scaling and carried out an asymptotic analysis to capture this instability in low wave-number limit. Depending on S and Bo, we observed the existence of a stable gap in terms of Ma where both the eigen-modes remain stable. Our results indicate that for a given Bond number, the width of stable gap in terms of Ma decreases with decrease in S and the stable gap vanishes when S is sufficiently small. The effect of increasing Bond number (or equivalently, the strength of basic flow) is found to be stabilizing for the film flow configuration.