Falling Liquid Films, 2012
Applied Mathematical Sciences Series, Vol. 176

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.

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 case: 3D flow. -7.1 Phenomenology. -7.2 2D modelling of 3D film flows. -7.3 Floquet Analysis: 3D stability of 2D waves. -7.4 2D simulations of 3D flows. -7.5 Salient features. -7.6 References and further reading. -7.7 Appendix. -8 Interaction of 3D solitary waves. -8.1 A model system for low-dimensional complexity. -8.2 Speed of 3D solitary pulses. -8.4 Approximate analytical solution. -8.5 Coherent structures theory. -8.6 Salient features. -8.7 References and further reading. -9 Heated films. -9.1 Formulation. -9.2 Formulation at first order. -9.3 Reduced models. -9.4 Regularized model. -9.5 Linear stability results. -9.6 Solitary waves. -9.7 Refined models. -9.8 Three-dimensional wave patterns. -9.9 Salient features. -9.10 References and further reading. -9.11 Appendix. -10 Reactive falling films. -10.1 Problem definition and governing equations. -10.2 Long-wave theory. -10.3 Kapita-Shkadov model and weighted residuals. -10.4 Salient features. -10.5 References and further reading. -10.6 Appendix. -11 Outlooks. -11.1 What has been offered/achieved. -11.2 Open questions – Suggestions. -11.3 References and further reading. -Conclusions. -Bibliography. -Index. -Summary.

Easy access to mathematical models of different degrees of complexity, capable of describing, with controllable accuracy, falling liquid films

Gives a detailed review of state-of-the-art theoretical, analytical and numerical methodologies

Provides a research-oriented exposition of the fundamentals of interfacial fluid mechanics, especially of its conceptual framework and the underlying assumptions

Includes supplementary material: sn.pub/extras