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 stability of a liquid flow bounded by slippery and porous walls is studied for infinitesimal disturbances of arbitrary wavenumbers, and 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.
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.
35 citations
TL;DR: Wei et al. as discussed by the authors studied the long-wave instability of a shear-imposed liquid flow down an inclined plane, where the free surface of the fluid is covered by an insoluble surfactant.
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.
33 citations
TL;DR: In this paper, 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.
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.
25 citations
TL;DR: In this paper, 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.
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.
19 citations
TL;DR: In this paper, an analysis of a pressure driven two-layer Poiseuille flow confined between a rigid wall and a Darcy-Brinkman porous layer is explored, and a linear stability analysis of the conservation laws leads to an Orr-Sommerfeld system to identify the time and length scales of the instabilities.
13 citations
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TL;DR: In this article, the effect of insoluble surfactant on the gravity-driven flow of a liquid film down an inclined wall with periodic undulations or indentations is investigated in the limit of vanishing Reynolds number.
Abstract: The effect of an insoluble surfactant on the gravity-driven flow of a liquid film down an inclined wall with periodic undulations or indentations is investigated in the limit of vanishing Reynolds number. A perturbation analysis for walls with small-amplitude sinusoidal corrugations reveals that the surfactant amplifies the deformation of the film surface, though it also renders the film thickness more uniform over the inclined surface. The effect of the surfactant is most significant when the film thickness is less than half the wall period. To explain the deforming influence of the surfactant, a linear stability analysis of film flow down an inclined plane is undertaken for two-dimensional perturbations. The results reveal the occurrence of a Marangoni normal mode whose rate of decay is lower than that of the single mode occurring in the absence of surfactants. Numerical methods based on a combined boundary-element/finite-volume method are implemented to compute flow down a periodic wall with large-amplitude corrugations or semi-circular depressions. In the case of a wavy wall, it is found that the shape of the film surface is described well by the linear perturbation expansion for small and moderate wave amplitudes. Streamline patterns reveal that, although the effect of the surfactant on the shape of the film surface is generally small, Marangoni tractions may have a profound influence on the kinematics by causing the onset of regions of recirculating flow.
70 citations
TL;DR: In this paper, a closed generalized momentum transport equation (GTE) is presented, which is expressed in terms of position-dependent effective transport coefficients, which are computed from the solution of associated closure problems.
Abstract: The momentum transfer between a homogeneous fluid and a porous medium in a system analogous to the one used by Beavers and Joseph (J Fluid Mech 30:197-207, 1967) is studied using volume averaging techniques. In this article, we present a closed generalized momentum transport equation (GTE) that is valid everywhere and is expressed in terms of position-dependent effective transport coefficients, which are computed from the solution of associated closure problems previously reported. A combination of the velocity profiles from the GTE in the definition of the excess terms that define the jump coefficients allows their computation using numerical techniques. The calculations are in concordance with those resulting from the work of Goyeau et al. (Int J Heat Mass Transf. 46:4071-4081, 2003), showing a strong dependence with the porosity. In addition, the effects of the roughness of the boundary on the computation of the position-dependent permeability tensor in the inter-region are also analyzed.
70 citations
TL;DR: In this paper, the authors derived the macroscopic momentum transport equation in a non-homogeneous solidifying columnar dendritic mushy zone using the method of volume averaging.
58 citations
TL;DR: Etude aux grands nombres de Reynolds de la stabilite lineaire bidimensionnelle d'ecoulements descendant sur un plan incline as mentioned in this paper, numerique des ondes de surface and de cisaillement suivant l'angle d'inclinaison, la tension superficielle and le facteur de forme
Abstract: Etude aux grands nombres de Reynolds de la stabilite lineaire bidimensionnelle d'ecoulements descendant sur un plan incline. Etude numerique des ondes de surface et de cisaillement suivant l'angle d'inclinaison, la tension superficielle et le facteur de forme
56 citations
TL;DR: In this paper, the effect of insoluble surfactant on linear stability of a shear-imposed flow down an inclined plane is examined in the long-wavelength limit, and the underlying mechanisms of the stability are also elucidated in detail.
Abstract: The effect of an insoluble surfactant on the linear stability of a shear-imposed flow down an inclined plane is examined in the long-wavelength limit. It has been known that a free falling film flow with surfactant is stable to long-wavelength disturbances at sufficiently small Reynolds numbers. Imposing an additional interfacial shear, however, could cause instability due to the shear-induced Marangoni effect. Two modes of the stability are identified and the corresponding growth rates are derived. The underlying mechanisms of the stability are also elucidated in detail.
54 citations