Thin film flow down a porous substrate in the presence of an insoluble surfactant: Stability analysis
TL;DR: In this paper, the stability of a gravity-driven film flow on a porous inclined substrate is considered, when the film is contaminated by an insoluble surfactant, in the frame work of Orr-Sommerfeld analysis.
Abstract: The stability of a gravity-driven film flow on a porous inclined substrate is considered, when the film is contaminated by an insoluble surfactant, in the frame work of Orr-Sommerfeld analysis. The classical long-wave asymptotic expansion for small wave numbers reveals the occurrence of two modes, the Yih mode and the Marangoni mode for a clean/a contaminated film over a porous substrate and this is confirmed by the numerical solution of the Orr-Sommerfeld system using the spectral-Tau collocation method. The results show that the Marangoni mode is always stable and dominates the Yih mode for small Reynolds numbers; as the Reynolds number increases, the growth rate of the Yih mode increases, until, an exchange of stability occurs, and after that the Yih mode dominates. The role of the surfactant is to increase the critical Reynolds number, indicating its stabilizing effect. The growth rate increases with an increase in permeability, in the region where the Yih mode dominates the Marangoni mode. Also, the growth rate is more for a film (both clean and contaminated) over a thicker porous layer than over a thinner one. From the neutral stability maps, it is observed that the critical Reynolds number decreases with an increase in permeability in the case of a thicker porous layer, both for a clean and a contaminated film over it. Further, the range of unstable wave number increases with an increase in the thickness of the porous layer. The film flow system is more unstable for a film over a thicker porous layer than over a thinner one. However, for small wave numbers, it is possible to find the range of values of the parameters characterizing the porous medium for which the film flow can be stabilized for both a clean film/a contaminated film as compared to such a film over an impermeable substrate; further, it is possible to enhance the instability of such a film flow system outside of this stability window, for appropriate choices of the porous substrate characteristics.
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TL;DR: In this article, the linear instability of a 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.
Abstract: 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 flow direction. A coupled Orr--Sommerfeld system of equations for the considered problem is derived using the perturbation technique and normal mode analysis. The eigenmodes corresponding to the Orr--Sommerfeld eigenvalue problem are obtained by employing the spectral collocation method. The numerical results imply that the stronger external shear destabilizes the interface mode instability. However, a stabilizing impact of the external shear on the surface mode is noticed if the shear is imposed in the flow direction, which is in contrast to the role of imposed external shear on the surface mode for a surfactant laden single layer falling film. Moreover, the impression of shear mode on the primary instability is analyzed in the high Reynolds number regime with sufficiently low inclination angle. Under such configuration, dominance of the shear mode over the surface mode is observed due to the weaker impact of the gravitation force on the surface instability. The shear mode can also be stabilized by applying the external shear in the counter direction of the streamwise flow. Conclusively, the extra imposed shear on the stratified two-layer falling film plays an active role to control the attitude of the instabilities.
8 citations
TL;DR: In this article, a linear stability analysis is performed in the presence of several flow parameters, and a coupled system of Orr-Sommerfeld equations is derived for the two-layer film flows with a free surface.
7 citations
Dissertation•
24 Sep 2014
TL;DR: In this article, the authors investigate the stabilite of fluides using electrocapillarite, a technique optique consistant a etudier la propagation and lattenuation d'ondes capillaires.
Abstract: Nous etudions la stabilite de l'ecoulement de fluide rheofluidifiant (pseudoplastique) sur plan incline. La connaissance des conditions d'apparition des instabilites interesse ici particulierement le secteur industriel faisant appel a des methodes de couchage (papeterie, photographie), ou le secteur environnemental dans la comprehension de certaines situations exceptionnelles (coulees de boues, laves torrentielles, ecoulements de glaciers). Nous modelisons la viscosite des fluides utilises par la loi de Carreau. Afin de caracteriser nos fluides, nous utilisons l'electrocapillarite comme technique optique consistant a etudier la propagation et l'attenuation d'ondes capillaires. Les resultats de mesures permettent en particulier de determiner la viscosite a valeur de cisaillement aussi faibles que 10−3s−1. Notre objectif est d'etudier experimentalement la stabilite de films rheofluidifiants sur plan incline. Pour des valeurs fixees de l'angle d'inclinaison, nous avons determine le seuil critique experimental et trace la courbe marginale de stabilite sur les plans (Re, k) et (Re, c) pour nos differents fluides. Nous trouvons que nos resultats experimentaux sont en bon accord avec les resultats numeriques, et confirment l'effet rheofluidifiant destabilisant relativement au cas Newtonien. Nous discutons enfin la validite du theoreme de Squire en ecrivant l'equation d'Orr-Sommerfeld generalisee aux ondes 3D et aux fluides de Carreau. Analytiquement, les relations de Squire ne sont pas verifiees, et les resultats numeriques montrent que les relations de Squire ne s'ecrivent que dans le cas Newtonien
6 citations
TL;DR: In this paper, the nonlinear dynamics of gravity-driven two-layer flow through an oblique microchannel of porous walls subjected to an external electric potential is examined, and the evolution equation governing the surface wave deflection is derived in the frame of long wave theory.
Abstract: The nonlinear dynamics of gravity-driven two-layers flow through an oblique microchannel of porous walls subjected to an external electric potential is examined. The evolution equation governing the surface wave deflection is derived in the frame of long wave theory. The stability criteria of the linearized system are investigated. As permeability, inclination or dielectric constant increase, the disturbances become stronger. However, the viscosity ratio plays an irregular role on the stability. Resonant waves propagating on the fluid interface are introduced. The instability of the base flow is simulated. It is observed that the instability onset can be controlled by many physical properties related with the model. The effect of permeability as well as dielectric constant corresponds to linear processing expectations. Viscosity ratio improves stability in certain situations. However, the electric role is generally dominant in the current model. Such results may be useful in practical applications by designing a device in order to control the instability. Solitary waves propagating on the interface are studied. The presence of stable stationary solitons is shown in certain statuses of the model.
6 citations
TL;DR: In this article , the influence of externally imposed shear on a surfactant-laden gravity-driven fluid flow over an inclined porous substrate is studied using the linear perturbation theory.
Abstract: The influence of externally imposed shear on a surfactant-laden gravity-driven fluid flow over an inclined porous substrate is studied using the linear perturbation theory. The hydrodynamic instability of the flow system corresponding to infinitesimal disturbances is examined in the framework of the Orr–Sommerfeld (OS) boundary value problem. Furthermore, the generalized OS model is obtained by including the Marangoni stress and external shear on the flow dynamics. The formulated stability problem is solved as an eigenvalue problem by the Chebyshev spectral collocation technique. The analysis encounters the existence of different classes of unstable modes, namely, the surface, surfactant, and shear modes. The surface mode instability occurs in the low range of Reynolds number and is the dominant mode of instability in particular parameter ranges. The imposed shear at the top surface along and opposite to the flow direction induces possible destabilization and stabilization of the flow, respectively. The permeability and porosity of the porous medium have a mixed impact on the surface mode instability. The temporal growth rate of the surface mode enhances for a thicker porous medium. The surface mode of the flow contaminated by an insoluble surfactant is less unstable than that of the clean free surface flow. This is due to the co-existence of the damped surfactant mode together with the unstable surface mode. On the other hand, the shear mode instability is identified at higher Reynolds numbers for a very small inclination angle, and the shear mode propagates faster for stronger imposed shear in the downstream direction. This trend is reversed for the upstream imposed shear. Moreover, the Marangoni effects exhibit the stabilizing influence on the shear mode. Conclusively, the external shear force would be helpful in regulating the instability of the surfactant-laden film flow down a porous medium.
5 citations
References
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Book•
01 Jan 1960
TL;DR: In this paper, the authors discuss the nature and properties of liquid interfaces, including the formation of a new phase, nucleation and crystal growth, and the contact angle of surfaces of solids.
Abstract: Capillarity. The Nature and Thermodynamics of Liquid Interfaces. Surface Films on Liquid Substrates. Electrical Aspects of Surface Chemistry. Long--Range Forces. Surfaces of Solids. Surfaces of Solids: Microscopy and Spectroscopy. The Formation of a New Phase--Nucleation and Crystal Growth. The Solid--Liquid Interface--Contact Angle. The Solid--Liquid Interface--Adsorption from Solution. Frication, Lubrication, and Adhesion. Wetting, Flotation, and Detergency. Emulsions, Foams, and Aerosols. Macromolecular Surface Films, Charged Films, and Langmuir--Blodgett Layers. The Solid--Gas Interface--General Considerations. Adsorption of Gases and Vapors on Solids. Chemisorption and Catalysis. Index.
10,790 citations
Book•
01 Jan 1992TL;DR: In this paper, an introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal.
Abstract: This introduction to convection in porous media assumes the reader is familiar with basic fluid mechanics and heat transfer, going on to cover insulation of buildings, energy storage and recovery, geothermal reservoirs, nuclear waste disposal, chemical reactor engineering and the storage of heat-generating materials like grain and coal. Geophysical applications range from the flow of groundwater around hot intrusions to the stability of snow against avalanches. The book is intended to be used as a reference, a tutorial work or a textbook for graduates.
5,570 citations
TL;DR: In this article, a simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.
Abstract: Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.
2,898 citations
TL;DR: In this article, a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations.
Abstract: Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.
2,689 citations
Book•
01 Jan 1999
TL;DR: In this article, a single-phase flow in homogeneous Porous Media is described. But the single phase flow is not a single phase of the Darcy's Law. But it is a phase of a single flow in a two-phase system.
Abstract: 1 Diffusion and Heterogeneous Reaction in Porous Media 2 Transient Heat Conduction in Two-Phase Systems 3 Dispersion in Porous Media 4 Single-Phase Flow in Homogeneous Porous Media: Darcy's Law 5 Single-Phase Flow in Heterogeneous Porous Media Appendix Nomenclature References Index
1,308 citations