Singularities: formation, structure, and propagation
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2015
|
| Schriftenreihe: | Cambridge texts in applied mathematics
|
| Schlagworte: | |
| Online-Zugang: | Contributor biographical information Publisher description Table of contents only Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xvi, 453 Seiten Illustrationen, Diagramme |
| ISBN: | 9781107098411 9781107485495 |
Internformat
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Datensatz im Suchindex
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|---|---|
| adam_text | Singularities: Formation, Structure,
and Propagation
J EGGERS
University of Bristol
M A FONTELOS
Consejo Superior de Investigaciones Cientificas, Madrid
ilf Cambridge
UNIVERSITY PRESS
Contents
Preface page xiii
PART I SETTING THE SCENE
1 What are singularities all about? 3
1 1 Drop pinch-off: scaling and universality 5
1 2 Stationary cusps: persistent singularities 9
1 3 Shock waves: propagation 10
Exercises ^ 12
2 Blowup 16
21A scalar example 16
2 2 Crossover 20
2 3 Regularization: saturation 22
2 4 What is special about power laws? 26
Exercises 28
3 Similarity profile 32
3 1 The spatial structure of blowup 32
3 2 Stability 39
3 3 Similarity solutions and the dynamical system 45
3 4 Regularization 47
3 5 Continuation 50
351 Similarity description 55
Exercises ’ 57
4 Continuum equations 63
4 1 General ideas 63
4 2 The Navier-Stokes equation 64
4 3 Boundary conditions 68
vii
Contents
viii
4 4 Free surface motion 70
4 5 Special limits 72
451 Potential flow 72
452 Two-dimensional flow 74
453 Stokes flow 76
4 6 felasticity 77
Exercises 82
5 Local singular expansions 89
5 1 Potential flow in a corner 89
5 2 Potential flow around a two-dimensional airfoil 93
5 3 Stokes waves 97
5 4 Electric fields near tips: Taylor cones 99
5 5 Mixed boundary conditions 102
5 6 Viscous flow in corners and Moffatt eddies 105
Exercises 110
6 Asymptotic expansions of PDEs 115
6 1 Thin film equation 115
611 Hele-Shaw flow 121
6 2 Slender jets 123
Exercises 129
PART II FORMATION OF SINGULARITIES
7 Drop breakup ^ 143
7 1 Overview and dimensional analysis 143
711 Surface tension-viscosity-inertia balance 144
712 Surface tension-inertia balance 145
713 Surface tension-viscosity balance 146
7 2 Viscous breakup 147
721 Lagrangian transformation 147
722 Similarity solutions 149
7 3 Generic breakup 154
731 The universal solution 156
732 Stability 164
7 4 Fluctuating jet equations 169
7 5 Inviseid breakup 172
7 6 Crossover 176
7 7 Fluid-fluid breakup 177
Exercises 182
Contents ix
8 A numerical example: drop pinch-off 186
8 1 Finite-difference scheme 186
8 2 Time stepping and stability 191
8 3 Grid refinement 198
8 4 Analysis of pinching 200
Exercises 203
9 Slow convergence 207
9 1 Mean curvature flow 207
9 2 Center-manifold analysis 209
9 3 Bubbles 214
931 Basics 214
932 Slender body theory 218
933 Cavity dynamics 220
934 Approach to the fixed point 223
Exercises 226
10 Continuation 230
10 1 Post-breakup solution: viscous thread 230
10 2 Regularization: thread formation for viscoelastic materials 239
10 2 1 Dilute polymer solutions 240
10 2 2 The beads-on-a-string configuration 243
10 3 Crossover: bubbles and satellites 249
Exercises 252
PART III PERSISTENT SINGULARITIES:
PROPAGATION
11 Shock waves 259
11 1 Burgers’ equation 259
11 2 Similarity description 264
11 3 Conservation laws: shocks and unique continuation 268
11 4 Viscosity solutions 272
11 5 Compressible gas flow 275
11 5 1 Unique continuation for systems 279
11 6 Imploding spherical shocks 285
11 6 1 Geometrical shock dynamics 289
Exercises 292
12 The dynamical system 298
12 1 Overview 298
12 2 Periodic orbits: a toy model 299
X
Contents
12 3 Discrete self-similarity in the implosion of
polygonal shocks 302
12 4 Chaos 307
Exercises 310
Vortices 313
13 1 Point vortices in inviscid fluid flow 316
13 1 1 Vortex motion 316
13 2 Vortex filaments 322
13 2 1 Comer singularity of a vortex filament 326
13 3 Vortex sheets 328
13 3 1 Linear instability of vortex sheets 333
13 3 2 Moore’s singularity of vortex sheets 335
13 3 3 Continuation of Moore’s singularity 339
13 4 Vortices in the Ginzburg-Landau equation 341
13 4 1 Structure of stationary vortices 343
13 4 2 The renormalized energy 346
13 4 3 Dynamics of Ginzburg-Landau
vortices 349
13 5 Nonlinear Schrödinger equation 352
Exercises 354
Cusps and caustics 358
14 1 Viscous free surface cusps 358
14 2 Singularity theory 364
14 3 Hele-Shaw flow 366
14 4 Optical caustics 372
14 5 The wavelength scale 378
Exercises 383
Contact lines and cracks 390
15 1 Driven singularities 390
15 2 A spreading drop 390
15 2 1 Voinov solution: the universal
singularity 394
15 2 2 Regularization: inner region 398
15 2 3 The drop: outer region 403
15 2 4 The global problem: matching 408
15 3 A moving crack 409
15 3 1 Universal tip singularity 410
15 3 2 Inner problem: the fracture energy 415
Contents
xi
15 3 3 The /-integral 417
15 3 4 The global problem 420
Exercises 423
Appendix A Vector calculus 427
Appendix B Index notation and the summation convention 431
Appendix C Dimensional analysis 434
References 436
|
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| spelling | Eggers, J. 1963- Verfasser (DE-588)1080528164 aut Singularities formation, structure, and propagation J. Eggers, University of Bristol, M.A. Fontelos, Consejo Superior de Investigaciones Cientificas, Madrid Cambridge Cambridge University Press 2015 xvi, 453 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Cambridge texts in applied mathematics Includes bibliographical references and index Singularities (Mathematics) Physik (DE-588)4045956-1 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Singularität Mathematik (DE-588)4077459-4 gnd rswk-swf Physik (DE-588)4045956-1 s Singularität Mathematik (DE-588)4077459-4 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Fontelos, Marco A. Verfasser (DE-588)140302093 aut http://www.loc.gov/catdir/enhancements/fy1512/2015016866-b.html Contributor biographical information http://www.loc.gov/catdir/enhancements/fy1512/2015016866-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy1512/2015016866-t.html Table of contents only HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028790343&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Eggers, J. 1963- Fontelos, Marco A. Singularities formation, structure, and propagation Singularities (Mathematics) Physik (DE-588)4045956-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Singularität Mathematik (DE-588)4077459-4 gnd |
| subject_GND | (DE-588)4045956-1 (DE-588)4044779-0 (DE-588)4077459-4 |
| title | Singularities formation, structure, and propagation |
| title_auth | Singularities formation, structure, and propagation |
| title_exact_search | Singularities formation, structure, and propagation |
| title_full | Singularities formation, structure, and propagation J. Eggers, University of Bristol, M.A. Fontelos, Consejo Superior de Investigaciones Cientificas, Madrid |
| title_fullStr | Singularities formation, structure, and propagation J. Eggers, University of Bristol, M.A. Fontelos, Consejo Superior de Investigaciones Cientificas, Madrid |
| title_full_unstemmed | Singularities formation, structure, and propagation J. Eggers, University of Bristol, M.A. Fontelos, Consejo Superior de Investigaciones Cientificas, Madrid |
| title_short | Singularities |
| title_sort | singularities formation structure and propagation |
| title_sub | formation, structure, and propagation |
| topic | Singularities (Mathematics) Physik (DE-588)4045956-1 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Singularität Mathematik (DE-588)4077459-4 gnd |
| topic_facet | Singularities (Mathematics) Physik Partielle Differentialgleichung Singularität Mathematik |
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