Linear stability of a contaminated fluid flow down a slippery inclined plane
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.
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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 article, a study of optimal temporal and spatial disturbance growths for three-dimensional viscous incompressible fluid flows with slippery walls was carried out under the framework of normal velocity and normal vorticity formulations, where a Chebyshev spectral collocation method was used to solve the governing equations numerically.
Abstract: A study of optimal temporal and spatial disturbance growths is carried out for three-dimensional viscous incompressible fluid flows with slippery walls. The non-modal temporal stability analysis is performed under the framework of normal velocity and normal vorticity formulations. A Chebyshev spectral collocation method is used to solve the governing equations numerically. For a free surface flow over a slippery inclined plane, the maximum temporal energy amplification intensifies with the effect of wall slip for the spanwise perturbation, but it attenuates with the wall slip when perturbation considers both streamwise and spanwise wavenumbers. It is found that the boundary for the regime of transient growth appears far ahead of the boundary for the regime of exponential growth, which raises a question on the critical Reynolds number for the shear mode predicted from the eigenvalue analysis. Furthermore, the eigenvalue analysis or the modal stability analysis reveals that the unstable region for the shear mode decays rapidly in the presence of wall slip, which is followed by the successive amplification of the critical Reynolds number for the shear mode and ensures the stabilizing effect of slip length on the shear mode. On the other hand, for a channel flow with slippery bounding walls, the maximum spatial energy amplification intensifies with the effect of wall slip in the absence of angular frequency, but it reduces with the wall slip if the angular frequency is present in the disturbance. Furthermore, the maximum spatial energy disturbance growth can be achieved if the disturbance excludes the angular frequency. Furthermore, it is observed that the angular frequency plays an essential role in the pattern formation of optimal response. In addition, the pseudo-resonance phenomenon occurs due to external temporal and spatially harmonic forcings, where the pseudo-resonance peak is much higher than the resonance peak.
19 citations
15 citations
TL;DR: In this article, the influence of thermal radiation on double-diffusive natural convection of a nanofluid past an inclined wavy surface in the presence of gyrotactic microorganisms is analyzed.
Abstract: In this article, the influence of thermal radiation on double-diffusive natural convection of a nanofluid past an inclined wavy surface in the presence of gyrotactic microorganisms is analysed. A coordinate transformation is used to transform the complex wavy surface to a smooth surface. Using the pseudo-similarity variables, the governing non-linear differential equations and their associate boundary stipulations are non-dimensionalized. The resulting system of non-linear partial differential equations is linearized using a local linearization method. The obtained linear system of partial differential equations is solved using the bivariate Chebyshev pseudo-spectral collocation method. The effect of pertinent physical and geometrical parameters on the physical quantities of the flow are illustrated through graphs and analyzed.
14 citations
TL;DR: In this paper, a linear stability analysis of a surfactant-laden viscoelastic liquid flowing down a slippery inclined plane is carried out under the framework of Orr-Sommerfeld type eigenvalue problem.
Abstract: A study of linear stability analysis of a surfactant-laden viscoelastic liquid flowing down a slippery inclined plane is carried out under the framework of Orr–Sommerfeld type eigenvalue problem. It is assumed that the viscoelastic liquid satisfies the rheological property of Walters' liquid B ″. The Orr–Sommerfeld type eigenvalue problem is solved analytically and numerically based on the long-wave analysis and Chebyshev spectral collocation method, respectively. The long-wave analysis predicts the existence of two temporal modes, the so-called surface mode and surfactant mode, where the first order temporal growth rate for the surfactant mode is zero. However, the first order temporal growth rate for the surface mode is non-zero, which leads to the critical Reynolds number for the surface mode. Further, it is found that the critical Reynolds number for the surface mode reduces with the increasing value of viscoelastic coefficient and ensures the destabilizing effect of viscoelastic coefficient on the primary instability induced by the surface mode in the long-wave regime. However, the numerical result demonstrates that the viscoelastic coefficient has a non-trivial stabilizing effect on the surface mode when the Reynolds number is far away from the onset of instability. Further, if the Reynolds number is high and the inclination angle is sufficiently low, there exists another mode, namely the shear mode. The unstable region induced by the shear mode magnifies significantly even for the weak effect of viscoelastic coefficient and makes the transition faster from stable to unstable flow configuration for the viscoelastic liquid. Moreover, the slip length exhibits a dual role in the surface mode as reported for the Newtonian liquid. But it exhibits only a stabilizing effect on the shear mode. In addition, it is found that the Marangoni number also exhibits a dual nature on the primary instability induced by the surface mode in contrast to the result of the Newtonian liquid.
11 citations
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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
Book•
28 Dec 2000
TL;DR: In this article, the authors present an approach to the Viscous Initial Value Problem with the objective of finding the optimal growth rate and the optimal response to the initial value problem.
Abstract: 1 Introduction and General Results.- 1.1 Introduction.- 1.2 Nonlinear Disturbance Equations.- 1.3 Definition of Stability and Critical Reynolds Numbers.- 1.3.1 Definition of Stability.- 1.3.2 Critical Reynolds Numbers.- 1.3.3 Spatial Evolution of Disturbances.- 1.4 The Reynolds-Orr Equation.- 1.4.1 Derivation of the Reynolds-Orr Equation.- 1.4.2 The Need for Linear Growth Mechanisms.- I Temporal Stability of Parallel Shear Flows.- 2 Linear Inviscid Analysis.- 2.1 Inviscid Linear Stability Equations.- 2.2 Modal Solutions.- 2.2.1 General Results.- 2.2.2 Dispersive Effects and Wave Packets.- 2.3 Initial Value Problem.- 2.3.1 The Inviscid Initial Value Problem.- 2.3.2 Laplace Transform Solution.- 2.3.3 Solutions to the Normal Vorticity Equation.- 2.3.4 Example: Couette Flow.- 2.3.5 Localized Disturbances.- 3 Eigensolutions to the Viscous Problem.- 3.1 Viscous Linear Stability Equations.- 3.1.1 The Velocity-Vorticity Formulation.- 3.1.2 The Orr-Sommerfeld and Squire Equations.- 3.1.3 Squire's Transformation and Squire's Theorem.- 3.1.4 Vector Modes.- 3.1.5 Pipe Flow.- 3.2 Spectra and Eigenfunctions.- 3.2.1 Discrete Spectrum.- 3.2.2 Neutral Curves.- 3.2.3 Continuous Spectrum.- 3.2.4 Asymptotic Results.- 3.3 Further Results on Spectra and Eigenfunctions.- 3.3.1 Adjoint Problem and Bi-Orthogonality Condition.- 3.3.2 Sensitivity of Eigenvalues.- 3.3.3 Pseudo-Eigenvalues.- 3.3.4 Bounds on Eigenvalues.- 3.3.5 Dispersive Effects and Wave Packets.- 4 The Viscous Initial Value Problem.- 4.1 The Viscous Initial Value Problem.- 4.1.1 Motivation.- 4.1.2 Derivation of the Disturbance Equations.- 4.1.3 Disturbance Measure.- 4.2 The Forced Squire Equation and Transient Growth.- 4.2.1 Eigenfunction Expansion.- 4.2.2 Blasius Boundary Layer Flow.- 4.3 The Complete Solution to the Initial Value Problem.- 4.3.1 Continuous Formulation.- 4.3.2 Discrete Formulation.- 4.4 Optimal Growth.- 4.4.1 The Matrix Exponential.- 4.4.2 Maximum Amplification.- 4.4.3 Optimal Disturbances.- 4.4.4 Reynolds Number Dependence of Optimal Growth.- 4.5 Optimal Response and Optimal Growth Rate.- 4.5.1 The Forced Problem and the Resolvent.- 4.5.2 Maximum Growth Rate.- 4.5.3 Response to Stochastic Excitation.- 4.6 Estimates of Growth.- 4.6.1 Bounds on Matrix Exponential.- 4.6.2 Conditions for No Growth.- 4.7 Localized Disturbances.- 4.7.1 Choice of Initial Disturbances.- 4.7.2 Examples.- 4.7.3 Asymptotic Behavior.- 5 Nonlinear Stability.- 5.1 Motivation.- 5.1.1 Introduction.- 5.1.2 A Model Problem.- 5.2 Nonlinear Initial Value Problem.- 5.2.1 The Velocity-Vorticity Equations.- 5.3 Weakly Nonlinear Expansion.- 5.3.1 Multiple-Scale Analysis.- 5.3.2 The Landau Equation.- 5.4 Three-Wave Interactions.- 5.4.1 Resonance Conditions.- 5.4.2 Derivation of a Dynamical System.- 5.4.3 Triad Interactions.- 5.5 Solutions to the Nonlinear Initial Value Problem.- 5.5.1 Formal Solutions to the Nonlinear Initial Value Problem.- 5.5.2 Weakly Nonlinear Solutions and the Center Manifold.- 5.5.3 Nonlinear Equilibrium States.- 5.5.4 Numerical Solutions for Localized Disturbances.- 5.6 Energy Theory.- 5.6.1 The Energy Stability Problem.- 5.6.2 Additional Constraints.- II Stability of Complex Flows and Transition.- 6 Temporal Stability of Complex Flows.- 6.1 Effect of Pressure Gradient and Crossflow.- 6.1.1 Falkner-Skan (FS) Boundary Layers.- 6.1.2 Falkner-Skan-Cooke (FSC) Boundary layers.- 6.2 Effect of Rotation and Curvature.- 6.2.1 Curved Channel Flow.- 6.2.2 Rotating Channel Flow.- 6.2.3 Combined Effect of Curvature and Rotation.- 6.3 Effect of Surface Tension.- 6.3.1 Water Table Flow.- 6.3.2 Energy and the Choice of Norm.- 6.3.3 Results.- 6.4 Stability of Unsteady Flow.- 6.4.1 Oscillatory Flow.- 6.4.2 Arbitrary Time Dependence.- 6.5 Effect of Compressibility.- 6.5.1 The Compressible Initial Value Problem.- 6.5.2 Inviscid Instabilities and Rayleigh's Criterion.- 6.5.3 Viscous Instability.- 6.5.4 Nonmodal Growth.- 7 Growth of Disturbances in Space.- 7.1 Spatial Eigenvalue Analysis.- 7.1.1 Introduction.- 7.1.2 Spatial Spectra.- 7.1.3 Gaster's Transformation.- 7.1.4 Harmonic Point Source.- 7.2 Absolute Instability.- 7.2.1 The Concept of Absolute Instability.- 7.2.2 Briggs' Method.- 7.2.3 The Cusp Map.- 7.2.4 Stability of a Two-Dimensional Wake.- 7.2.5 Stability of Rotating Disk Flow.- 7.3 Spatial Initial Value Problem.- 7.3.1 Primitive Variable Formulation.- 7.3.2 Solution of the Spatial Initial Value Problem.- 7.3.3 The Vibrating Ribbon Problem.- 7.4 Nonparallel Effects.- 7.4.1 Asymptotic Methods.- 7.4.2 Parabolic Equations for Steady Disturbances.- 7.4.3 Parabolized Stability Equations (PSE).- 7.4.4 Spatial Optimal Disturbances.- 7.4.5 Global Instability.- 7.5 Nonlinear Effects.- 7.5.1 Nonlinear Wave Interactions.- 7.5.2 Nonlinear Parabolized Stability Equations.- 7.5.3 Examples.- 7.6 Disturbance Environment and Receptivity.- 7.6.1 Introduction.- 7.6.2 Nonlocalized and Localized Receptivity.- 7.6.3 An Adjoint Approach to Receptivity.- 7.6.4 Receptivity Using Parabolic Evolution Equations.- 8 Secondary Instability.- 8.1 Introduction.- 8.2 Secondary Instability of Two-Dimensional Waves.- 8.2.1 Derivation of the Equations.- 8.2.2 Numerical Results.- 8.2.3 Elliptical Instability.- 8.3 Secondary Instability of Vortices and Streaks.- 8.3.1 Governing Equations.- 8.3.2 Examples of Secondary Instability of Streaks and Vortices.- 8.4 Eckhaus Instability.- 8.4.1 Secondary Instability of Parallel Flows.- 8.4.2 Parabolic Equations for Spatial Eckhaus Instability.- 9 Transition to Turbulence.- 9.1 Transition Scenarios and Thresholds.- 9.1.1 Introduction.- 9.1.2 Three Transition Scenarios.- 9.1.3 The Most Likely Transition Scenario.- 9.1.4 Conclusions.- 9.2 Breakdown of Two-Dimensional Waves.- 9.2.1 The Zero Pressure Gradient Boundary Layer.- 9.2.2 Breakdown of Mixing Layers.- 9.3 Streak Breakdown.- 9.3.1 Streaks Forced by Blowing or Suction.- 9.3.2 Freestream Turbulence.- 9.4 Oblique Transition.- 9.4.1 Experiments and Simulations in Blasius Flow.- 9.4.2 Transition in a Separation Bubble.- 9.4.3 Compressible Oblique Transition.- 9.5 Transition of Vortex-Dominated Flows.- 9.5.1 Transition in Flows with Curvature.- 9.5.2 Direct Numerical Simulations of Secondary Instability of Crossflow Vortices.- 9.5.3 Experimental Investigations of Breakdown of Cross-flow Vortices.- 9.6 Breakdown of Localized Disturbances.- 9.6.1 Experimental Results for Boundary Layers.- 9.6.2 Direct Numerical Simulations in Boundary Layers.- 9.7 Transition Modeling.- 9.7.1 Low-Dimensional Models of Subcritical Transition.- 9.7.2 Traditional Transition Prediction Models.- 9.7.3 Transition Prediction Models Based on Nonmodal Growth.- 9.7.4 Nonlinear Transition Modeling.- III Appendix.- A Numerical Issues and Computer Programs.- A.1 Global versus Local Methods.- A.2 Runge-Kutta Methods.- A.3 Chebyshev Expansions.- A.4 Infinite Domain and Continuous Spectrum.- A.5 Chebyshev Discretization of the Orr-Sommerfeld Equation.- A.6 MATLAB Codes for Hydrodynamic Stability Calculations.- A.7 Eigenvalues of Parallel Shear Flows.- B Resonances and Degeneracies.- B.1 Resonances and Degeneracies.- B.2 Orr-Sommerfeld-Squire Resonance.- C Adjoint of the Linearized Boundary Layer Equation.- C.1 Adjoint of the Linearized Boundary Layer Equation.- D Selected Problems on Part I.
2,215 citations
TL;DR: In this article, the Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm.
Abstract: The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.
1,365 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
TL;DR: A review of experimental studies regarding the phenomenon of slip of Newtonian liquids at solid interfaces is provided in this article, with particular attention to the effects that factors such as surface roughness, wettability and the presence of gaseous layers might have on the measured interfacial slip.
Abstract: For several centuries fluid dynamics studies have relied upon the assumption that when a liquid flows over a solid surface, the liquid molecules adjacent to the solid are stationary relative to the solid. This no-slip boundary condition (BC) has been applied successfully to model many macroscopic experiments, but has no microscopic justification. In recent years there has been an increased interest in determining the appropriate BCs for the flow of Newtonian liquids in confined geometries, partly due to exciting developments in the fields of microfluidic and microelectromechanical devices and partly because new and more sophisticated measurement techniques are now available. An increasing number of research groups now dedicate great attention to the study of the flow of liquids at solid interfaces, and as a result a large number of experimental, computational and theoretical studies have appeared in the literature. We provide here a review of experimental studies regarding the phenomenon of slip of Newtonian liquids at solid interfaces. We dedicate particular attention to the effects that factors such as surface roughness, wettability and the presence of gaseous layers might have on the measured interfacial slip. We also discuss how future studies might improve our understanding of hydrodynamic BCs and enable us to actively control liquid slip.
985 citations