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.

Texte du rabat

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.

Résumé

This monograph provides a detailed review of the 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. 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 experiments 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 the underlying basic physics.

The book will be of particular interest 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; researchers working on both applied and fundamental theoretical and experimental aspects of thin film flows; and engineers and technologists dealing with processes involving thin films, either isothermal or heated.

Topics covered include:

Detailed derivations of governing equations and wall and free-surface boundary conditions for free-surface thin film flows in the presence of thermocapillary Marangoni effect; linear stability including Orr-Sommerfeld, absolute/convective instability and Floquet analysis of periodic waves; strongly nonlinear analysis including construction of bifurcation diagrams of periodic and solitary waves; weakly nonlinear prototypes such as Kuramoto-Sivashinsky equation; validity domain of the long-wave expansions; kinematic/dynamic waves, connection with shallow water and river flows/hydraulic jumps; dynamical systems approach, local and global bifurcations, homoclinicity and conditions for periodic, subsidiary and secondary homoclinic orbits; modulation instability of solitary waves to transverse perturbations; transition to two-dimensional solitary waves and interaction of two-dimensional solitary waves; and substrate heating and competition between solitary waves and rivulet formation in free-surface flows over heated substrates.

Tutorials and details of computational methodologies including computer programs:

Solution of the Orr-Sommerfeld eigenvalue problem; computational search via continuation for traveling wave solutions and their bifurcations; computation of systems of nonlinear pde's using finite differences; spectral representation and aliasing.

Contenu

Preface. -Acknowledgements. -Nomenclature. -Introduction. -Review of Phenomenology. -Modelling: state-of-the-art. -Structure of the book. -1 Flow and heat transfer. -1.1 Governing equations and boundary conditions. -1.2 Dimensionless equations, scalings and parameters. -1.3 The role of the Biot number. -1.4 Salient features. -1.5 References and further reading. -2 Primary instability. -2.1 Linear stability analysis. -2.2 Transverse disturbances. -2.3 Longtitudinal disturbances. -2.4 Mechanism of the hydrodynamic instability. -2.5 Salient features. -2.6 References and further reading. -3 Boundary layer-like approximation. -3.1 Boundary layer equations. -3.2 2D Boundary Layer Equations. -3.3 Strong surface tension limit. -3.4 Shkadov's scaling. -3.5 Reduction of the governing equations. -3.7 Scalings: three sets of parameters. -3.8 Salient features. -3.9 References and further reading. -3.10 Appendix. -4 Methodologies for flows at low Re. -4.1 Long-wave asymptotic expansion. -4.2 Validity domain of the Benney equation. -4.3 Parametic study for closed and open flows. -4.4 Regularization a la Pade. -4.5 Comparison of the different one-equation models. -4.6 Weakly nonlinear models. -4.7 Salient features. -4.8 References and further reading. -4.9 Appendix. -4.10 Physical parameters. -4.11 Small Biot number, analogy with forced convection. -5 Methodologies for moderate Re. -5.1 Averaged two-equation models. -5.2 Relaxing the self-similar assumption. -5.3 Methods of weighted residuals. -5.4 First-order formulation. -5.5 Comparison of methods of weighted residuals. -5.6 Second-order formulation -5.7 Reduction of the full second order model. -5.8 Salient features. -5.9 References and further reading. -6 Isothermal case: 2D flow. -6.1 Linear stability analysis. -6.2 Travelling waves. -6.3 Spatial evolution of 2D waves. -6.4 Salient features. -6.5 References and further reading. -6.6 Appendix. -7 Isothermal …