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

Cover......Page 1
Pattern Formation at Interfaces (Springer, 2010)......Page 4
ISBN 978-3-7091-0124-7......Page 5
Preface......Page 6
Table of Contents......Page 8
Contents......Page 9
1 Introduction......Page 11
2.1 Physical mechanisms of patterns and waves......Page 14
2.2 Application-oriented aspects......Page 17
2.3 Dimensionless numbers and time scales......Page 19
2.4 Other instability mechanisms in very thin liquid films......Page 21
3 Basic equations and boundary conditions......Page 23
3.1 Non-negligible gas thermal conductivity......Page 26
3.2 Generalized one-sided modeling of evaporation......Page 27
3.3 Reference states......Page 30
3.4 Linear stability analysis......Page 32
Monotonic modes......Page 34
Oscillatory modes......Page 36
3.5 Direct numerical simulations......Page 43
4.1 The Swift-Hohenberg equation and its variants......Page 45
4.2 Basic symmetries of Bénard set-ups......Page 47
4.3 Symmetries and amplitude equations......Page 49
Bifurcation of rolls......Page 50
Bifurcation of hexagonal patterns......Page 51
4.4 Long-wave order-parameter equations for patterns......Page 55
5 Acknowledgments......Page 61
Bibliography......Page 62
Contents......Page 65
1 Introduction......Page 67
2 Reaction-diffusion systems......Page 69
Animals’ disease......Page 72
3.1 Stationary solutions......Page 73
3.2 Front solutions......Page 75
3.3 Motion of the front edge......Page 76
3.4 Non-generic fronts......Page 78
4.1 Fronts between locally stable phases......Page 79
4.2 Lyapunov functional......Page 81
4.3 Allen-Cahn equation......Page 83
4.4 Interaction between kinks......Page 85
4.5 Phase transition in an external field......Page 88
4.6 Domain wall pinned by an inhomogeneity......Page 90
4.7 Curved fronts of the phase transition......Page 93
5.1 Formulation of the problem......Page 96
5.2 Plane stationary front......Page 98
5.3 Dynamics of curved fronts......Page 101
5.4 Linear stability theory of the planar front......Page 105
Monotonic instability......Page 106
Oscillatory instability......Page 107
Monotonic longwave instability......Page 108
Oscillatory shortwave instability......Page 110
Bibliography......Page 111
Contents......Page 112
1.2 Instabilities......Page 114
1.3 Pattern formation – Examples......Page 118
1.4 Types of instabilities......Page 120
Decomposition of the velocity field, basic equations......Page 124
B. Surface of the fluid......Page 126
2.2 Linear stability analysis......Page 127
2.3 Numerics......Page 128
The method......Page 129
Stability......Page 130
Closed upper surface......Page 132
3 Thick films with undeformable surface – binary mixtures......Page 133
3.1 The basic equations and boundary conditions......Page 134
3.3 Nonlinear solutions......Page 137
4.1 Order parameters......Page 140
4.2 The Ginzburg-Landau equation......Page 141
4.3 The Swift-Hohenberg equation......Page 144
Gradient expansion......Page 145
Stripes, hexagons, and squares......Page 146
5 Thin films with a deformable surface......Page 151
5.1 Reduced two-dimensional description – perfect fluids......Page 152
The shallow water equations......Page 153
Numerical solutions......Page 155
The lubrication approximation......Page 156
Laplace pressure and gravity......Page 158
The disjoining pressure and ultra-thin films......Page 161
Normal form......Page 163
Numerical solutions......Page 164
Physical values......Page 166
Thin film equation and parameters......Page 168
The disjoining pressure......Page 169
Fluid parameters......Page 170
Holes or drops?......Page 171
Normal Form......Page 173
Results: the horizontal layer......Page 174
6.4 The inclined layer......Page 177
Bibliography......Page 178
Contents......Page 180
1.2 Disjoining Pressure......Page 182
1.3 Effective Mobility......Page 183
1.4 Contact Angle......Page 184
2.1 Perturbation Expansion......Page 186
2.2 Translational Solvability Condition......Page 187
2.3 Motion due to Asymmetry of Contact Angles......Page 189
3.1 Moving Droplet on a Precursor Film......Page 190
3.2 Mass Transport through the Precursor......Page 192
3.3 Coarsening......Page 193
3.4 Migration of Interacting Droplets......Page 195
4.1 Substrate Modification......Page 196
4.2 Traveling Bifurcation......Page 200
4.3 Non-diffusive Limit......Page 201
4.5 Scattering......Page 204
5.1 Static Thickness Fronts......Page 208
5.2 Evaporation and Condensation......Page 210
5.3 Fluxes and Mobility of the Front......Page 212
5.4 Solvability Condition......Page 213
6.1 Straight-line front......Page 215
6.2 “Pancake” and “hole”......Page 217
6.3 Solution in a comoving frame......Page 219
6.4 Zigzag instability......Page 221
Bibliography......Page 224
Contents......Page 225
1 Equilibrium crystal shape......Page 227
2 Growth of a spherical crystal nucleus......Page 233
3 Mullins-Sekerka instability......Page 238
4 Dendrites......Page 240
5 Surface diffusion and surface-diffusion-controlled interface shape......Page 245
6 Elastic instability of solid epitaxial films and self-assembly of quantum dots......Page 251
Bibliography......Page 258
Contents......Page 260
1.1 The Physics of the Instability......Page 262
1.2 The Model......Page 267
1.3 The Base Solution......Page 270
1.4 The Perturbation Equations......Page 271
2.2 The physical model......Page 277
2.3 Physics of the phase-change problem without convection......Page 278
2.4 Physics of the phase-change problem with convection......Page 279
2.5 The mathematical model......Page 280
2.6 The base state solution and the perturbed equations......Page 283
Results of calculations and explanation......Page 286
2.7 Stabilizing effect of the vapor flow......Page 292
3.1 Introduction......Page 295
3.3 Mathematical model......Page 296
3.5 Perturbation expansions......Page 299
3.6 Overview......Page 300
3.7 Perturbed equations......Page 302
3.8 Discussion......Page 308