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E-grāmata: Singularities: Formation, Structure, and Propagation

(University of Bristol), (Universidad Autónoma de Madrid)
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Singularities: Formation, Structure, and PropagationPalielināt
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Many natural phenomena are described as singularities, for example, the formation of drops and bubbles, or the motion of cracks. Aimed at a broad audience of students and researchers in mathematics, physics and engineering, this book provides mathematical tools for understanding all aspects of singularities.

Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more specialised methods of partial differential equations, complex analysis and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations.

Recenzijas

'The book will serve as an excellent introduction to the field of singularities in continuum mechanics, and a valuable resource for researchers In short, a wonderful achievement!' H. K. Moffatt, Journal of Fluid Mechanics

Papildus informācija

This book explores a wide range of singular phenomena, providing mathematical tools for understanding them and highlighting their common features.
Preface xiii
PART I SETTING THE SCENE
1 What are singularities all about?
3(13)
1.1 Drop pinch-off: scaling and universality
5(4)
1.2 Stationary cusps: persistent singularities
9(1)
1.3 Shock waves: propagation
10(6)
Exercises
12(4)
2 Blowup
16(16)
2.1 A scalar example
16(4)
2.2 Crossover
20(2)
2.3 Regularization: saturation
22(4)
2.4 What is special about power laws?
26(6)
Exercises
28(4)
3 Similarity profile
32(31)
3.1 The spatial structure of blowup
32(7)
3.2 Stability
39(6)
3.3 Similarity solutions and the dynamical system
45(2)
3.4 Regularization
47(3)
3.5 Continuation
50(13)
3.5.1 Similarity description
55(2)
Exercises
57(6)
4 Continuum equations
63(26)
4.1 General ideas
63(1)
4.2 The Navier--Stokes equation
64(4)
4.3 Boundary conditions
68(2)
4.4 Free surface motion
70(2)
4.5 Special limits
72(5)
4.5.1 Potential flow
72(2)
4.5.2 Two-dimensional flow
74(2)
4.5.3 Stokes flow
76(1)
4.6 Elasticity
77(12)
Exercises
82(7)
5 Local singular expansions
89(26)
5.1 Potential flow in a corner
89(4)
5.2 Potential flow around a two-dimensional airfoil
93(4)
5.3 Stokes waves
97(2)
5.4 Electric fields near tips: Taylor cones
99(3)
5.5 Mixed boundary conditions
102(3)
5.6 Viscous flow in corners and Moffatt eddies
105(10)
Exercises
110(5)
6 Asymptotic expansions of PDEs
115(28)
6.1 Thin film equation
115(8)
6.1.1 Hele-Shaw flow
121(2)
6.2 Slender jets
123(20)
Exercises
129(14)
PART II FORMATION OF SINGULARITIES
7 Drop breakup
143(43)
7.1 Overview and dimensional analysis
143(4)
7.1.1 Surface tension--viscosity--inertia balance
144(1)
7.1.2 Surface tension--inertia balance
145(1)
7.1.3 Surface tension--viscosity balance
146(1)
7.2 Viscous breakup
147(7)
7.2.1 Lagrangian transformation
147(2)
7.2.2 Similarity solutions
149(5)
7.3 Generic breakup
154(15)
7.3.1 The universal solution
156(8)
7.3.2 Stability
164(5)
7.4 Fluctuating jet equations
169(3)
7.5 Inviscid breakup
172(4)
7.6 Crossover
176(1)
7.7 Fluid--fluid breakup
177(9)
Exercises
182(4)
8 A numerical example: drop pinch-off
186(21)
8.1 Finite-difference scheme
186(5)
8.2 Time stepping and stability
191(7)
8.3 Grid refinement
198(2)
8.4 Analysis of pinching
200(7)
Exercises
203(4)
9 Slow convergence
207(23)
9.1 Mean curvature How
207(2)
9.2 Center-manifold analysis
209(5)
9.3 Bubbles
214(16)
9.3.1 Basics
214(4)
9.3.2 Slender body theory
218(2)
9.3.3 Cavity dynamics
220(3)
9.3.4 Approach to the fixed point
223(3)
Exercises
226(4)
10 Continuation
230(29)
10.1 Post-breakup solution: viscous thread
230(9)
10.2 Regularization: thread formation for viscoelastic materials
239(10)
10.2.1 Dilute polymer solutions
240(3)
10.2.2 The beads-on-a-string configuration
243(6)
10.3 Crossover: bubbles and satellites
249(10)
Exercises
252(7)
PART III PERSISTENT SINGULARITIES: PROPAGATION
11 Shock waves
259(39)
11.1 Burgers' equation
259(5)
11.2 Similarity description
264(4)
11.3 Conservation laws: shocks and unique continuation
268(4)
11.4 Viscosity solutions
272(3)
11.5 Compressible gas flow
275(10)
11.5.1 Unique continuation for systems
279(6)
11.6 Imploding spherical shocks
285(13)
11.6.1 Geometrical shock dynamics
289(3)
Exercises
292(6)
12 The dynamical system
298(15)
12.1 Overview
298(1)
12.2 Periodic orbits: a toy model
299(3)
12.3 Discrete self-similarity in the implosion of polygonal shocks
302(5)
12.4 Chaos
307(6)
Exercises
310(3)
13 Vortices
313(45)
13.1 Point vortices in inviscid fluid flow
316(6)
13.1.1 Vortex motion
316(6)
13.2 Vortex filaments
322(6)
13.2.1 Coiner singularity of a vortex filament
326(2)
13.3 Vortex sheets
328(13)
13.3.1 Linear instability of vortex sheets
333(2)
13.3.2 Moore's singularity of vortex sheets
335(4)
13.3.3 Continuation of Moore's singularity
339(2)
13.4 Vortices in the Ginzburg--Landau equation
341(11)
13.4.1 Structure of stationary vortices
343(3)
13.4.2 The renormalized energy
346(3)
13.4.3 Dynamics of Ginzburg--Landau vortices
349(3)
13.5 Nonlinear Schrodinger equation
352(6)
Exercises
354(4)
14 Cusps and caustics
358(32)
14.1 Viscous free surface cusps
358(6)
14.2 Singularity theory
364(2)
14.3 Hele-Shaw flow
366(6)
14.4 Optical caustics
372(6)
14.5 The wavelength scale
378(12)
Exercises
383(7)
15 Contact lines and cracks
390(37)
15.1 Driven singularities
390(1)
15.2 A spreading drop
390(19)
15.2.1 Voinov solution: the universal singularity
394(4)
15.2.2 Regularization: inner region
398(5)
15.2.3 The drop: outer region
403(5)
15.2.4 The global problem: matching
408(1)
15.3 A moving crack
409(18)
15.3.1 Universal tip singularity
410(5)
15.3.2 Inner problem: the fracture energy
415(2)
15.3.3 The J-integral
417(3)
15.3.4 The global problem
420(3)
Exercises
423(4)
Appendix A Vector calculus 427(4)
Appendix B Index notation and the summation convention 431(3)
Appendix C Dimensional analysis 434(2)
References 436(10)
Index 446
J. Eggers is Professor of Applied Mathematics at the University of Bristol. His career has been devoted to the understanding of self-similar phenomena, and he has more than fifteen years of experience in teaching non-linear and scaling phenomena to undergraduate and postgraduate students. Eggers has made fundamental contributions to our mathematical understanding of free-surface flows, in particular the break-up and coalescence of drops. His work was instrumental in establishing the study of singularities as a research field in applied mathematics and in fluid mechanics. He is a member of the Academy of Arts and Sciences in Erfurt, Germany, a fellow of the American Physical Society, and has recently been made a Euromech Fellow. M. A. Fontelos is a researcher in applied mathematics at the Spanish Research Council (CSIC). His scientific work has focused on partial differential equations and their applications to fluid mechanics, with special emphasis on the study of singularities and free-surface flows. His main results concern the formation of singularities (or not) combining the use of rigorous mathematical results with asymptotic and numerical methods.
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