Pattern Formation at Interfaces (CISM International Centre for Mechanical Sciences, 513)

✍ Scribed by Pierre Colinet (editor), Alexander Nepomnyashchy (editor)

• Publisher: Springer
• Year: 2010
• Tongue: English
• Leaves: 309
• Category: Library

✦ Synopsis

The book deals with modern methods of nonlinear stability theory applied to problems of continuous media mechanics in the presence of interfaces, with applications to materials science, chemical engineering, heat transfer technologies, as well as in combustion and other reaction-diffusion systems. Interfaces play a dominant role at small scales, and their correct modeling is therefore also crucial in the rapidly expanding fields of microfluidics and nanotechnologies. To this aim, the book combines contributions of eminent specialists in the field, with a special emphasis on rigorous and predictive approaches. Other goals of this volume are to allow the reader to identify key problems of high scientific value, and to see the similarity between a variety of seemingly different physical problems.

✦ Table of Contents

Title Page
Copyright Page
Preface
Table of Contents
Interfacial patterns and waves in liquid layers and thin films
Contents
1 Introduction
2 Phenomenology of Bénard instabilities
2.1 Physical mechanisms of patterns and waves
2.2 Application-oriented aspects
2.3 Dimensionless numbers and time scales
2.4 Other instability mechanisms in very thin liquid films
3 Basic equations and boundary conditions
3.1 Non-negligible gas thermal conductivity
3.2 Generalized one-sided modeling of evaporation
3.3 Reference states
3.4 Linear stability analysis
Monotonic modes
Oscillatory modes
3.5 Direct numerical simulations
4 Simplified models for non-equilibrium patterns
4.1 The Swift-Hohenberg equation and its variants
4.2 Basic symmetries of Bénard set-ups
4.3 Symmetries and amplitude equations
Bifurcation of rolls
Bifurcation of hexagonal patterns
4.4 Long-wave order-parameter equations for patterns
5 Acknowledgments
Bibliography
Nonlinear dynamics of fronts
Contents
1 Introduction
2 Reaction-diffusion systems
Calcium waves in eggs of fish and amphibiae
Chemotaxis
Animals’ disease
3 The Fisher-Kolmogorov equation
3.1 Stationary solutions
3.2 Front solutions
3.3 Motion of the front edge
3.4 Non-generic fronts
4 Multistability
4.1 Fronts between locally stable phases
4.2 Lyapunov functional
4.3 Allen-Cahn equation
4.4 Interaction between kinks
4.5 Phase transition in an external field
4.6 Domain wall pinned by an inhomogeneity
4.7 Curved fronts of the phase transition
5 Combustion fronts
5.1 Formulation of the problem
5.2 Plane stationary front
5.3 Dynamics of curved fronts
5.4 Linear stability theory of the planar front
Monotonic instability
Oscillatory instability
5.5 Nonlinear development of front instabilities
Monotonic longwave instability
Oscillatory shortwave instability
Bibliography
Three Dimensional Film Dynamics
Contents
1 Introduction
1.1 Basic equations
1.2 Instabilities
1.3 Pattern formation – Examples
1.4 Types of instabilities
2 Thick films with undeformable surface – Pure fluids
2.1 The basic equations
Scaling to non-dimensional quantities
Decomposition of the velocity field, basic equations
Boundary conditions
A. Bottom Plate
B. Surface of the fluid
2.2 Linear stability analysis
2.3 Numerics
The method
Stability
Accuracy
2.4 Results
Closed upper surface
Free upper surface
3 Thick films with undeformable surface – binary mixtures
3.1 The basic equations and boundary conditions
3.2 The linear problem – codimension-two point
3.3 Nonlinear solutions
4 Reduced description – Order parameter equations
4.1 Order parameters
4.2 The Ginzburg-Landau equation
4.3 The Swift-Hohenberg equation
Gradient expansion
Stripes, hexagons, and squares
5 Thin films with a deformable surface
5.1 Reduced two-dimensional description – perfect fluids
The shallow water equations
Numerical solutions
5.2 Reduced two-dimensional description – viscous fluids
The lubrication approximation
Laplace pressure and gravity
The disjoining pressure and ultra-thin films
6 Spinodal dewetting
6.1 Ultra-thin isothermal films
Normal form
Numerical solutions
Metastable region and nucleation
Physical values
6.2 Externally heated thin films
Thin film equation and parameters
The disjoining pressure
Fluid parameters
Holes or drops?
6.3 Time dependent numerical solutions
Normal Form
Results: the horizontal layer
6.4 The inclined layer
Bibliography
Thin Film and Droplet Patterns Shaped by Surface Forces
Contents
1 Basic Equations
1.1 Evolution Equation
1.2 Disjoining Pressure
1.3 Effective Mobility
1.4 Contact Angle
2 Quasistationary Motion
2.1 Perturbation Expansion
2.2 Translational Solvability Condition
2.3 Motion due to Asymmetry of Contact Angles
3 Interactions Mediated by the Precursor
3.1 Moving Droplet on a Precursor Film
3.2 Mass Transport through the Precursor
3.3 Coarsening
3.4 Migration of Interacting Droplets
4 Chemical Self-Propulsion
4.1 Substrate Modification
4.2 Traveling Bifurcation
4.3 Non-diffusive Limit
4.4 Relaxation to a Stationary Pattern
4.5 Scattering
5 Thickness Fronts
5.1 Static Thickness Fronts
5.2 Evaporation and Condensation
5.3 Fluxes and Mobility of the Front
5.4 Solvability Condition
6 Evaporative Patterns
6.1 Straight-line front
6.2 “Pancake” and “hole”
6.3 Solution in a comoving frame
6.4 Zigzag instability
Acknowledgement
Bibliography
Interfacial Phenomena in Materials Science
Contents
1 Equilibrium crystal shape
2 Growth of a spherical crystal nucleus
3 Mullins-Sekerka instability
4 Dendrites
5 Surface diffusion and surface-diffusion-controlled interface shape
6 Elastic instability of solid epitaxial films and self-assembly of quantum dots
Bibliography
The Physics and Analyses of Interfacial Instabilities that Arise from Phase Change
Contents
1 The Physics and Analysis of Instability During the Solidification of a Pure Material
1.1 The Physics of the Instability
1.2 The Model
1.3 The Base Solution
1.4 The Perturbation Equations
2 Evaporative Instability - Linear Theory
2.1 Introduction
2.2 The physical model
2.3 Physics of the phase-change problem without convection
2.4 Physics of the phase-change problem with convection
2.5 The mathematical model
2.6 The base state solution and the perturbed equations
Results of calculations and explanation
2.7 Stabilizing effect of the vapor flow
3 Evaporative instability in bilayer systems - Nonlinear theory
3.1 Introduction
3.2 The physical model
3.3 Mathematical model
3.4 Nonlinear analysis
3.5 Perturbation expansions
3.6 Overview
3.7 Perturbed equations
3.8 Discussion