| Description |
1 online resource (xv, 440 pages) |
| Series |
Applied mathematical sciences, 0066-5452 ; v. 176 |
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Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 176
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| Contents |
880-01 Introduction -- Flow and heat transfer -- Primary instability -- Boundary layer approximation -- Methodologies for low Re flows -- Methodologies for moderate Re flows -- Isothermal case: 2D flow -- Isothermal case: 3D flow -- Interaction of 3D solitary waves -- Heated films -- Reactive films -- Open questions and suggestions for further research |
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880-01/(S Machine generated contents note: 1.1. Brief Historical Perspective -- 1.2. Basic Phenomena of a Falling Film -- 1.2.1. Surface Wave Instability -- 1.2.2. Flow Evolution Features -- 1.2.3. Marangoni Effect -- 1.2.4. Inhomogeneous Heating -- 1.3. Mathematical Modeling and Methodologies -- 1.4. Structure and Contents of the Monograph -- 2.1. Governing Equations and Boundary Conditions -- 2.2. Dimensionless Equations, Scalings and Parameters -- 2.3. On the Development of the Nusselt Flat Film Solution -- 2.4. On the Two Wall Thermal Boundary Conditions: Retrieving ST from HF -- 2.5. Role of the Biot Number -- 3.1. Linearized Equations for the Disturbances -- 3.2. Orr-Sommerfeld Eigenvalue Problem -- 3.3. Oscillatory Versus Stationary (or Monotonic) Instabilities -- 3.4. Transverse Perturbations: kx = 0, kz = k -- 3.4.1. Eigenvalue Problem -- 3.4.2. Neutral Stability Condition -- 3.4.3. Critical Condition and Long-Wave Expansion -- 3.5. Streamwise Perturbations: kx = k, kz = 0 -- 3.5.1. Eigenvalue Problem -- 3.5.2. Neutral Stability Condition -- 3.5.3. Long-Wave Expansion -- 3.5.4. Critical Condition -- 3.5.5. Higher Order in the Long-Wave Expansion of the Dispersion Relation -- 3.6. Mechanism of the Hydrodynamic Instability -- 3.6.1. Energy Balance of the Perturbation -- 3.6.2. Vorticity Balance at the Perturbed Interface -- 3.6.3. Summary of the Key Factors for the Hydrodynamic Instability -- 4.1. Three-Dimensional Boundary Layer Equations -- 4.2. Two-Dimensional Boundary Layer Equations -- 4.3. On the Significance of the Second-Order Contributions -- 4.4. Strong Surface Tension Limit -- 4.5. Dissipation -- 4.6. Shkadov Scaling -- 4.7. Use of the Shkadov Scaling to Analyze the Balance of Different Forces on a Solitary Pulse -- 4.7.1. (i) Large-Amplitude Waves -- 4.7.2. (ii) Small-Amplitude Waves -- 4.7.3. Behavior of the Eigenvalues of the Flat Film Solution of a Linearized Averaged Model -- 4.8. Cross-stream Inertia -- 4.8.1. On the Order of Magnitude of Cross-stream Inertia -- 4.8.2. On the Region of Validity of the Boundary Layer Approximation -- 4.9. Reduction of the Boundary Layer Equations -- 4.9.1. Drag-Gravity and Drag-Inertia Regimes -- 4.9.2. Hierarchy of Models -- 4.10. Scalings: Three Sets of Parameters -- 5.1. Long-Wave Theory -- 5.1.1. Evolution Equation for the Film Thickness -- 5.1.2. Higher-Order Terms in the Gradient Expansion -- 5.1.3. Primary Instability for the First-Order BE -- 5.1.4. On the Relative Order Between the Wavenumber of Interfacial Disturbances and the Gradient Expansion Parameter -- 5.1.5. Comparison with Orr-Sommerfeld -- 5.2. Weakly Nonlinear Models -- 5.2.1. Models in Two Dimensions -- 5.2.2. Models in Three Dimensions -- 5.3. Traveling Waves -- 5.3.1. Closed and Open Flow Conditions -- 5.3.2. Traveling Wave Solutions in the Drag-Gravity Regime -- 5.4. Validity Domain of the BE -- 5.4.1. Blow up Versus Wavenumber -- 5.5. Parametric Study for Closed and Open Flows -- 5.5.1. BE with the Shkadov Scaling -- 5.5.2. Isothermal Vertical Films: Closed and Open Flows -- 5.5.3. Influence of Inclination -- 5.5.4. Influence of the Marangoni Effect in the Small Biot Number Limit: B " 1 -- 5.5.5. Subcritical Behavior of the BE -- 5.5.6. Concluding Remarks on the Validity Domain of the BE -- 5.6. Regularization with Padé Approximants -- 5.7. Generalization of the Single-Equation Model Obtained with Regularization -- 6.1. Background and Motivation -- 6.2. Averaged Two-Equation Models -- 6.2.1. Kapitza-Shkadov Model -- 6.2.2. Higher-Level Models Based on the Self-similar Closure -- 6.3. Center Manifold Analysis -- 6.4. Relaxing the Self-similar Assumption -- 6.5. Method of Weighted Residuals -- 6.6. First-Order Formulation -- 6.7. Comparison of Weighted Residuals Methods -- 6.7.1. Method of Subdomains -- 6.7.2. Collocation Method -- 6.7.3. Integral-Collocation Method -- 6.7.4. Method of Moments -- 6.7.5. Galerkin Method -- 6.7.6. Remarks -- 6.8. Second-Order Formulation -- 6.8.1. Full Second-Order Model -- 6.8.2. Simplified Second-Order Model -- 6.9. Reduction of the Full Second-Order Model -- 6.9.1. Elimination of s1 and s2 -- 6.9.2. Padé-Like Regularization -- 6.10. Contrasting the Center Manifold Analysis and the Method of Weighted Residuals -- 7.1. Linear Stability Analysis -- 7.1.1. Dispersion Relations and Neutral Stability -- 7.1.2. Absolute and Convective Instabilities -- 7.1.3. Wave Hierarchy -- 7.2. Traveling Waves -- 7.2.1. Dynamical Systems Approach -- 7.2.2. Solitary Wave Characteristics for δ " 1 and δ " 1 -- 7.2.3. Closed Flow Conditions -- 7.2.4. Open Flow Conditions -- 7.3. Spatio-temporal Evolution of Two-Dimensional Waves -- 7.3.1. Periodic Forcing -- 7.3.2. Noise-Driven Flows -- 8.1. Phenomena -- 8.2. Modeling of Three-Dimensional Film Flows -- 8.3. Floquet Analysis: Three-Dimensional Stability of γ1 Waves -- 8.4. Simulations of Three-Dimensional Flows -- 8.4.1. Three-Dimensional Modulations of γ1 Waves -- 8.4.2. Three-Dimensional Modulations of γ2 Waves -- 8.4.3. Three-Dimensional Natural (Noise-Driven) Waves -- 9.1. Formulation -- 9.2. Formulation at First Order -- 9.3. Formulation at Second Order -- 9.4. Reduced Models -- 9.4.1. Gradient Expansion -- 9.4.2. Reduction of the Full Second-Order Model -- 9.4.3. Padé-Like Regularization -- 9.5. Linear Stability -- 9.5.1. Neutral Stability Curves -- 9.5.2. Growth Rate Curves -- 9.5.3. Influence of Bi, Ma, Pr, Γ on the Neutral Stability Curves -- 9.5.4. Influence of Inclination -- 9.6. Solitary Waves -- 9.6.1. Bifurcation Diagrams -- 9.6.2. Drag-Gravity Regime -- 9.6.3. Drag-Inertia Regime -- 9.6.4. Limitations Related to the Surface Temperature Equation -- 9.7. Three-Dimensional Regularized Model -- 9.7.1. Small-Size Domain -- 9.7.2. Large-Size Domain -- A.1. Piotr Leonidovitch Kapitza (1894-1984) -- A.2. Carlo Giuseppe Matteo Marangoni (1840-1925) -- B.1. Surface Tension Relation -- B.2. Newton's Law of Cooling -- C.1. Heat Flux Boundary Condition (HF) -- C.2. Surface Gradient Operator -- C.3. On the Choice of the Unit Vectors Tangential to the Surface -- C.4. On the Evaluation of the Right Hand Side of the Tangential Stress Balance (2.13) -- C.5. Short Library of Weakly Nonlinear Model Equations: Bottom-up Dispersion Relation Approach -- C.6. Negative Polarity in the BKdV Equation -- C.7. Padé Approximants -- C.8. Center Manifold Projection for a Scalar Equation -- D.1. Viscous-Gravity Scaling -- D.2. On the Orders of Magnitude for the Different Groups in the Boundary Layer Equations -- D.3. Dimensionless Groups and Their Relationships for the ST Case -- D.4. Physical Parameters -- E.1. Dynamical System Corresponding to the Full Second-Order Model -- E.2. Three-Dimensional Full Second-Order Model -- E.3. Three-Dimensional Regularized Second-Order Model -- E.4. Full Second-Order Model for the ST Case -- E.5. Second-Order Inertia Corrections to the Regularized Model (9.33a), (9.33b) for the ST Case -- E.6. Weighted Residuals Modeling for the HF Case -- E.6.1. Formulation at First-Order -- E.6.2. Formulation at Second-Order -- F.1. Solving the Orr-Sommerfeld Equation by Continuation -- F.1.1. AUTO Source Code -- F.2. Computational Search for Traveling Wave Solutions and Their Bifurcations -- F.2.1. Hopf Bifurcation -- F.2.2. Period-Doubling Bifurcation -- F.2.3. Locus of Saddle-Node Bifurcation Points -- F.2.4. AUTO Source Code -- F.3. Time-Dependent Computations Using Finite Differences -- F.4. Spectral Representation and Aliasing |
| Summary |
Falling Liquid Films gives a detailed review of state-of-the-art theoretical, analytical and numerical methodologies, for the analysis of dissipative wave dynamics and pattern formation on the surface of a film falling down a planar inclined substrate. This prototype is an open-flow hydrodynamic instability, that represents an excellent paradigm for the study of complexity in active nonlinear media with energy supply, dissipation and dispersion. It will also be of use for a more general understanding of specific events characterizing the transition to spatio-temporal chaos and weak/dissipative turbulence.¡ Particular emphasis is given to low-dimensional approximations for such flows through a hierarchy of modeling approaches, including equations of the boundary-layer type, averaged formulations based on weighted residuals approaches and long-wave expansions. Whenever possible the link between theory and experiment is illustrated, and, as a further bridge between the two, the development of order-of-magnitude estimates and scaling arguments is used to facilitate the understanding of basic, underlying physics. ¡ This monograph will appeal to advanced graduate students in applied mathematics, science or engineering undertaking research on interfacial fluid mechanics or studying fluid mechanics as part of their program. It will also be of use to researchers working on both applied, fundamental theoretical and experimental aspects of thin film flows, as well as engineers and technologists dealing with processes involving isothermal or heated films. This monograph is largely self-contained and no background on interfacial fluid mechanics is assumed |
| Analysis |
Mathematics |
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Visualization |
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Engineering mathematics |
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Applications of Mathematics |
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Fluid- and Aerodynamics |
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Theoretical, Mathematical and Computational Physics |
| Bibliography |
Includes bibliographical references and index |
| Subject |
Liquid films -- Mathematical models
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Hydrodynamics -- Mathematical models
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Fluid dynamics.
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Mathematics.
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Hydrodynamics
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Mathematics
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mathematics.
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applied mathematics.
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SCIENCE -- Energy.
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SCIENCE -- Mechanics -- General.
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SCIENCE -- Physics -- General.
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Mathematics
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Fluid dynamics
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Hydrodynamics -- Mathematical models
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| Form |
Electronic book
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| Author |
Kalliadasis, Serafim
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| ISBN |
9781848823679 |
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1848823673 |
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