Thin film flow down a porous substrate in the presence of an insoluble surfactant: Stability analysis

Role of slip on the linear stability of a liquid flow through a porous channel

Abstract: The linear stability of a liquid flow bounded by slippery and porous walls is studied for infinitesimal disturbances of arbitrary wavenumbers. The Orr-Sommerfeld type eigenvalue problem is formulated by using the normal mode decomposition and resolved based on the Chebyshev spectral collocation method along with the QZ algorithm. The results are computed numerically in detail for various values of the flow parameters. The presence of an upper wall slip shows a destabilizing effect on the fluid layer mode, but it shows a stabilizing effect on the porous layer mode. On the other hand, the decreasing value of the depth ratio has a stabilizing effect on the fluid layer mode but it has a destabilizing effect on the porous layer mode. In fact, there occurs a competition between the most unstable porous layer mode and the most unstable fluid layer mode to control the primary instability. The most unstable porous layer mode triggers the primary instability unless the upper wall slip dominates the effect of the porous layer otherwise the most unstable fluid layer mode triggers the primary instability. A new phase boundary is detected in the plane of the depth ratio and slip length, which separates the domain of the most unstable porous layer mode from the domain of the most unstable fluid layer mode.

Linear stability analysis of a surfactant-laden shear-imposed falling film

Abstract: A study of the linear stability analysis of a shear-imposed fluid flowing down an inclined plane is performed when the free surface of the fluid is covered by an insoluble surfactant. The purpose is to extend the earlier work [H. H. Wei, “Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane,” Phys. Fluids 17, 012103 (2005)] for disturbances of arbitrary wavenumbers. The Orr-Sommerfeld boundary value problem is formulated and solved numerically based on the Chebyshev spectral collocation method. Two temporal modes, the so-called surface mode and surfactant mode, are detected in the long-wave regime. The surfactant mode becomes unstable when the Peclet number exceeds its critical value. In fact, the instability of the surfactant mode occurs on account for the imposed shear stress. Energy budget analysis predicts that the kinetic energy of the infinitesimal disturbance grows with the imposed shear stress. On the other hand, the numerical results reveal that both surface and surfactant modes can be destabilized by increasing the value of the imposed shear stress. Similarly, it is demonstrated that the shear mode becomes more unstable in the presence of the imposed shear stress. However, it can be stabilized by incorporating the insoluble surfactant at the free surface. Apparently, it seems that inertia does not play any role in the surfactant mode in the moderate Reynolds number regime. Furthermore, the competition between surface and shear modes is discussed.

Linear stability of a contaminated fluid flow down a slippery inclined plane

Abstract: The linear stability analysis of a fluid flow down a slippery inclined plane is carried out when the free surface of the fluid is contaminated by a monolayer of insoluble surfactant. The aim is to extend the earlier study [Samanta et al., J. Fluid Mech. 684, 353 (2011)] for low to high values of the Reynolds number in the presence of an insoluble surfactant. The Orr-Sommerfeld equation (OSE) is derived for infinitesimal disturbances of arbitrary wave numbers. At low Reynolds number, the OSE is solved analytically by using the long-wave analysis, which shows that the critical Reynolds number decreases in the presence of a slippery plane but increases in the presence of an insoluble surfactant. This fact ensures a destabilizing effect of wall slip and a stabilizing effect of insoluble surfactant on the long-wave surface mode. Further, the Chebyshev spectral collocation method is implemented to tackle the OSE equation numerically for an arbitrary value of the Reynolds number, or equivalently, for an arbitrary value of the wave number. At moderate Reynolds number, wall slip exhibits a stabilizing effect on the surface mode as opposed to the result in the long-wave regime, while the insoluble surfactant exhibits a stabilizing effect on the surface mode as in the result of the long-wave regime. On the other hand, at high Reynolds number, both wall slip and insoluble surfactant exhibit a stabilizing effect on the shear mode. Further, it is shown that both surface and shear modes compete with each other to dominate the primary instability once the inclination angle is sufficiently small. In addition, new phase boundaries are identified to differentiate the regimes of surface and shear modes.

Instabilities in viscosity-stratified two-fluid channel flow over an anisotropic-inhomogeneous porous bottom

Abstract: A linear stability analysis of a pressure driven, incompressible, fully developed laminar Poiseuille flow of immiscible two-fluids of stratified viscosity and density in a horizontal channel bounded by a porous bottom supported by a rigid wall, with anisotropic and inhomogeneous permeability, and a rigid top is examined. The generalized Darcy model is used to describe the flow in the porous medium with the Beavers-Joseph condition at the liquid-porous interface. The formulation is within the framework of modified Orr-Sommerfeld analysis, and the resulting coupled eigenvalue problem is numerically solved using a spectral collocation method. A detailed parametric study has revealed the different active and coexisting unstable modes: porous mode (manifests as a minimum in the neutral boundary in the long wave regime), interface mode (triggered by viscosity-stratification across the liquid-liquid interface), fluid layer mode [existing in moderate or O(1) wave numbers], and shear mode at high Reynolds numbers. As a result, there is not only competition for dominance among the modes but also coalescence of the modes in some parameter regimes. In this study, the features of instability due to two-dimensional disturbances of porous and interface modes in isodense fluids are explored. The stability features are highly influenced by the directional and spatial variations in permeability for different depth ratios of the porous medium, permeability and ratio of thickness of the fluid layers, and viscosity-stratification. The two layer flow in a rigid channel which is stable to long waves when a highly viscous fluid occupies a thicker lower layer can become unstable at higher permeability (porous mode) to long waves in a channel with a homogeneous and isotropic/anisotropic porous bottom and a rigid top. The critical Reynolds number for the dominant unstable mode exhibits a nonmonotonic behaviour with respect to depth ratio. However, it increases with an increase in anisotropy parameter ξ indicating its stabilizing role. Switching of dominance of modes which arises due to variations in inhomogeneity of the porous medium is dependent on the permeability and the depth ratio. Inhomogeneity arising due to an increase in vertical variations in permeability renders short wave modes to become more unstable by enlarging the unstable region. This is in contrast to the anisotropic modulations causing stabilization by both increasing the critical Reynolds number and shrinking the unstable region. A decrease in viscosity-stratification of isodense fluids makes the configuration hosting a less viscous fluid in a thinner lower layer adjacent to a homogeneous, isotropic porous bottom to be more unstable than the one hosting a highly viscous fluid in a thicker lower layer. An increase in relative volumetric flow rate results in switching the dominant mode from the interface to fluid layer mode. It is evident from the results that it is possible to exercise more control on the stability characteristics of a two-fluid system overlying a porous medium in a confined channel by manipulating the various parameters governing the flow configurations. This feature can be effectively exploited in relevant applications by enhancing/suppressing instability where it is desirable/undesirable.

Instabilities of a confined two-layer flow on a porous medium: An Orr–Sommerfeld analysis

Abstract: Instabilities of a pressure driven two-layer Poiseuille flow confined between a rigid wall and a Darcy–Brinkman porous layer are explored. A linear stability analysis of the conservation laws leads to an Orr–Sommerfeld system, which is solved numerically with appropriate boundary conditions to identify the time and length scales of the instabilities.fde The study uncovers the coexistence of twin instability modes, (i) long-wave interfacial mode—engendered by the viscosity stratification across the interface and (ii) finite wave number shear mode—originating from the inertial stresses, for almost all combinations of the viscosity ( μ r ) and thickness ( h r ) ratios of the liquid layers. The presence of the porous layer reduces the frictional influence on the films, which significantly alters the length and time scales of the shear mode while the interfacial mode remains dormant to this effect. This is in stark contrast to the two-layer flow confined between non-porous plates where an unstable interfacial (shear) mode is observed when μ r > h r 2 ( μ r h r 2 ) . The study reveals that strength of the shear mode, (a) increases with porosity, (b) initially increases and then becomes constant with porous layer thickness, (c) initially increases then reduces with increase in permeability, and (d) reduce with increase in the stress jump coefficient at the porous–liquid interface. Moreover, the gravity expedites the destabilization of both the modes in the inclined channels as compared to the similar non-inclined channels. The parametric study presented can find important applications in enhancing heat and mass transfer, mixing, and emulsification especially in the microscale flows.

Physical chemistry of surfaces

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.

Convection in Porous Media

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.

Boundary conditions at a naturally permeable wall

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.

Long-scale evolution of thin liquid films

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.

The method of volume averaging

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