Evolution of spontaneous structures in dissipative continuous systems

Lecture Notes in Physics: Monographs 55 ......Page 1
Evolution of Spontaneous Structures in Dissipative Continuous Systems ......Page 3
Copyright ......Page 4
Preface ......Page 5
Contents ......Page 8
1.1 Overview ......Page 10
1.2 Example: A Reaction Diffusion Equation ......Page 13
1.3 Linear Stability Analysis ......Page 14
1.4 Nonlinear Approaches ......Page 18
2 Symmetries and Symmetry Groups ......Page 19
3.1 Stationary Bifurcations ......Page 22
3.2 Hopf Bifurcation ......Page 27
3.3 Generalizations ......Page 33
3.4 Normal Form Computations ......Page 36
4 Generic Bifurcations for Lattice Symmetries ......Page 38
4.1 Rectangular Lattice ......Page 39
4.2 Square Lattice ......Page 42
4.3 Hexagonal Lattice ......Page 46
5 Further Developments ......Page 51
6.1 Beyond Finite-Dimensional Bifurcation Analysis ......Page 56
6.2 Construction of the GL Equations in 1D ......Page 57
6.3 Remarks ......Page 59
6.4 Example ......Page 62
6.5 Construction of GL Equations in 2D ......Page 64
7.1 Stationary Bifurcations, omega_c = o ......Page 67
7.2 The Complex GL Equation (CGLE) ......Page 72
8.1 Complications and Generalisations ......Page 83
8.2 Phase Equations ......Page 85
8.3 Final Remarks ......Page 87
References ......Page 88
1 Introduction ......Page 95
2.1 Historical Remarks ......Page 96
2.2 Geometry of the flow ......Page 97
2.3 Wavy Taylor vortex flow ......Page 98
2.4 Localised oscillations ......Page 99
2.5 Mode coupling ......Page 101
2.6 Period doubling scenarios ......Page 102
2.7 Homoclinic bifurcations and chaos ......Page 104
2.9 Phase dynamics ......Page 106
3.1 Theoretical Concepts ......Page 110
3.2 Experimental Results ......Page 112
3.3 The Influence of Azimuthal Modulations ......Page 115
4 Time Dependent Convection in Two Superimposed Fluid Layers ......Page 117
5.1 Introduction ......Page 121
5.2 Theoretical and numerical work ......Page 124
5.3 Experimental observations of thermocapillary flow structures in cylindrical liquid bridges ......Page 126
5.4 Thermocapillary flow structures in rotationally symmetric liquid layers – annular layers ......Page 128
References ......Page 130
Pattern Formation in Binary Fluid Convection and in Systems with Through fiow ......Page 136
1 Introduction ......Page 137
2 Extended 2D convection structures in fluid mixtures ......Page 139
2.1 The system ......Page 140
2.2 Bifurcation behavior ......Page 142
2.3 Structure of TW and SOC states ......Page 145
2.4 Appearance of bistable TWs with increasing Soret coupling ......Page 150
2.6 Universal scaling of TW frequencies and mixing properties ......Page 152
2.7 Growth dynamics ......Page 153
2.8 Models ......Page 157
3.1 Bifurcation behavior ......Page 163
3.2 Spatiotemporal structure ......Page 164
4 Dufour effect in gas mixtures ......Page 169
4.1 Stability analysis of the basic state ......Page 170
4.2 Nonlinear SOC and TW properties ......Page 172
5 3D convection structures in fluid mixtures: squares, crossrolls, and a subharmonic bifurcation cascade ......Page 173
5.1 Full-scale 3D numerical simulations ......Page 174
5.3 Oscillations between x-rolls, squares, and y-rolls ......Page 175
5.4 Subharrnonic bifurcation cascade to stationary crossrolls ......Page 177
6 The effect of throughflow on supercritically bifurcating patterns ......Page 178
6.1 Absolute and convective instability ......Page 179
6.2 Pattern selection in the absolutely unstable regime: a nonlinear eigenvalue problem ......Page 181
6.3 Noise sustained pattern growth in the convectively unstable regime: bulk versus boundary effects ......Page 186
7 Influence of throughflow on binary fluid convection ......Page 191
7.1 Linear properties ......Page 192
7.2 Nonlinear TW^U, TW^D, and TW^S states ......Page 197
Acknowledgments ......Page 199
References ......Page 200
1 Introduction ......Page 206
2.1 Wavelets ......Page 210
2.2 Archetypes ......Page 211
2.3 Proper Orthogonal Decomposition ......Page 213
3.1 Introduction ......Page 216
3.2 Data Base ......Page 217
3.3 Comparison of Coherent Structures – Prerequisites ......Page 219
3.4 Comparison of Coherent Structures – Results ......Page 221
3.5 Evolution of Spatial Complexity ......Page 224
3.6 Comparison of Coherent Structures – Conclusions ......Page 225
4 POD-Galerkin Models for the Dynamics of Coherent Structures ......Page 226
4.1 The Dynamical Equations ......Page 227
4.2 Instantaneous Energy Flows ......Page 230
5 Other Applications and Related Work ......Page 233
6 Discussion ......Page 235
References ......Page 236
1 Introduction ......Page 239
2 Basic equations ......Page 240
3.1 The eigenvalue problem ......Page 243
3.2 Ideal fluids and the limit of weak dissipation ......Page 244
3.3 Rigorous treatment of viscous dissipation ......Page 246
3.4 Analytic stability analysis in the limit of weak dissipation ......Page 247
3.5 Comparison with experiments ......Page 250
4 Nonlinear dynamics in finite geometries ......Page 251
5.1 Theoretical background ......Page 257
5.2 Ordered patterns at single-frequency drive ......Page 258
5.3 Ordered patterns at multi-frequency drive ......Page 260
5.5 Amplitude equations ......Page 263
5.6 Quasi one-dimensional waves: Annulus experiments ......Page 265
5.7 Instabilities of square patterns and the transition to spatio-temporal chaos ......Page 266
5.8 Model equations ......Page 269
6 Summary, open problems, perspectives ......Page 271
References ......Page 273
1 Introduction ......Page 275
2 Porous media convection (PMC) ......Page 277
3 Taylor-Couette Flow with Random Spatial Variations ......Page 283
4 Simple Models for Inhomogeneities in pattern formation ......Page 288
4.1 Swift-Hohenberg with spatially varying control parameter ......Page 289
4.2 Effects of spatially varying drifts ......Page 292
4.3 Modulated Rayleigh Bénard Convection ......Page 296
4.4 Concluding remarks ......Page 299
References ......Page 300
1 Introduction ......Page 304
2 Static Instabilities ......Page 307
2.1 Theoretical Formulation ......Page 308
2.2 Frederiks Transition ......Page 310
2.3 Undulational Instability ......Page 312
2.4 Interaction between Instabilities ......Page 314
2.5 Buckling Instability ......Page 316
2.6 Polarization Frederiks Transition ......Page 320
3 Electroconvection Theory in Smectic C and Smectic C* Films ......Page 325
3.1 Linear Equations ......Page 326
3.3 Symmetries and Floquet Analysis ......Page 327
3.4 Onset of Convection for DC Voltage ......Page 329
3.5 Onset of Convection for AC Driving Voltage in SmC* Films ......Page 330
3.6 Codimension-3 Bifurcation ......Page 332
3.7 Parameter Dependencies ......Page 333
4.1 General ......Page 335
4.3 Experiments on Freely Suspended Smectic C Films ......Page 336
4.4 Electroconvection in smectic C* films ......Page 337
References ......Page 340
1 Introduction ......Page 344
2 Basic equations and instability mechanisms ......Page 346
3 Elements of the Theoretical Analysis ......Page 349
4.1 General background ......Page 352
4.2 The weak electrolyte model (WEM) ......Page 354
4.3 Abnormal rolls ......Page 358
4.4 Dielectric regime ......Page 362
5 EHC in the homeotropic configuration ......Page 365
6.1 Experimental situation and motivation ......Page 370
6.2 Modelling of stochastically driven EHC ......Page 373
6.3 Stochastic stability criteria ......Page 375
6.4 Results ......Page 377
6.5 Concluding remarks and outlook ......Page 383
Acknowledgements ......Page 385
References ......Page 386
1 Introduction ......Page 393
2.1 Structural transformations on Pt surfaces ......Page 394
2.2 Reconstruction :,rnodel of CO oxidation on Pt(110) ......Page 395
2.3 Backfiring, spiral breakup and turbulence ......Page 396
3.1 Pattern formation in the liquid phase and on surfaces ......Page 397
3.2 Rectangular patterns and wave fragments in the NO + H2 reaction on Rh(110) ......Page 400
3.3 Triangular wave patterns ......Page 404
3.4 Mechanistic aspects and formulation of realistic models ......Page 406
4 Global coupling ......Page 408
4.1 Complex Ginzburg-Landau equation (CGLE) with global coupling ......Page 409
4.2 Globally coupled reconstruction model for CO oxidation on Pt (110) ......Page 412
5 Summarry ......Page 416
References ......Page 417
1 Introduction ......Page 420
2.1 Stationary Turing patterns ......Page 422
2.2 Excitation waves ......Page 424
2.3 Three-dimensional topologies ......Page 427
2.4 Waves and patterns in heterogeneous systems ......Page 429
3.1 Chemical preparation and observation method ......Page 431
3.2 The observed flow behavior ......Page 433
4 The coupling mechanism between reaction and convection ......Page 437
5 Chemically driven convection – a simple model ......Page 438
6 Chemically driven convection can stabilize Turing Patterns ......Page 439
7 Oscillatory hydrodynamic flow due to concentration dependence of surface tension ......Page 445
References ......Page 452
1 Introduction ......Page 455
2 Simulation of current filaments in n-GaAs ......Page 458
2.1 Monte Carlo simulation of impact ionization breakdow-n ......Page 459
2.2 Current filament formation in point contact samples ......Page 462
2.3 Current filaments in Corbino disks ......Page 469
2.4 Spatio-temporal spiking ......Page 472
3 Visualization of current filaments in n-GaAs ......Page 473
3.1 Experimental techniques ......Page 474
3.2 Current fillaments for various contact geometries ......Page 475
3.3 Lateral excursions of filaments in a magnetic field ......Page 479
4.1 Experimental details ......Page 484
4.3 Results ......Page 486
5 Self-organization in thin ZnS: Mn films ......Page 492
5.1 Experimental set-up ......Page 493
5.2 Results and Discussion ......Page 494
6 Conclusions ......Page 496
References ......Page 499
1 Introduction ......Page 504
3 Dynarnics of Granular Flow ......Page 505
3.1 Fluidization of Granular Matter ......Page 506
3.2 Segregation of Granular Materials ......Page 514
3.4 Acknowledgement ......Page 546
Appendix: Numerical methods for simulating the dynamics of granular materials ......Page 547
References ......Page 550
1 Introduction ......Page 555
2 The Disk Dynamo ......Page 556
3 A Homogeneous Dynamo ......Page 558
4 An experimental honogeneous dynamo ......Page 560
5 Numerical simulations ......Page 564
6 Non-linear effects ......Page 566
References ......Page 567
1 Introduction ......Page 568
2 The life cycle ......Page 569
3 cAMP signalling system ......Page 573
4 Chemotactic cell movement ......Page 576
5 Aggregation ......Page 578
6 Mound stage of development ......Page 580
7 Slug stage of development ......Page 585
8 Culmination ......Page 587
9 Conclusions and outlook ......Page 588
References ......Page 589