Seven internationally respected contributors provide a coherent discussion of an important area in theoretical physics - an ideal book for researchers.
This book presents five sets of lectures by internationally respected researchers on nonlinear instabilities and the transition to turbulence in hydrodynamics. The book begins with a general introduction to hydrodynamics covering fluid properties, flow measurement, dimensional analysis, and turbulence. Chapter Two reviews the special characteristics of instabilities in open flows. Chapter Three presents mathematical tools for multiscale analysis and asymptotic matching applied to the dynamics of fronts and localized nonlinear states. Chapter Four gives a detailed review of pattern forming instabilities. The final chapter provides a detailed and comprehensive introduction to the instability of flames, shocks and detonations. Together, these lectures provide a thought-provoking overview of current work in this important area for researchers in condensed matter physics and applied mathematics.
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' important reading for students embarking on a research career in hydrodynamics In summary, I very much enjoyed reading this book, and learned much from it I expect that it will find a wide readership among theoretical hydrodynamicists, and will still be useful well into the next century.' M. R. E. Proctor, European Journal of Mechanics
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Seven internationally respected contributors provide a coherent discussion of an important area in theoretical physics - an ideal book for researchers.
Preface xiii(3) Contributors xvi Overview 1(24) P. Manneville 1 An introduction to hydrodynamics 25(56) B. Castaing 1 What is a fluid? 25(15) 1.1 Introduction 25(1) 1.2 Viscosity and Reynolds number 26(2) 1.3 The basic equations 28(1) 1.4 Momentum budget: two examples 29(3) 1.5 Energy and entropy budget 32(2) 1.6 The Navier-Stokes equation 34(1) 1.7 Life at low Reynolds numbers 35(5) 2 The mechanisms 40(17) 2.1 Vorticity: diffusion, freezing, sources 40(4) 2.2 Energy transfer in turbulence 44(4) 2.3 Boundary layers and their separation 48(6) 2.4 Compressibility 54(3) 3 Flow measurement methods 57(5) 3.1 Introduction 57(1) 3.2 Hot wire anemometer 58(3) 3.3 Doppler laser anemometer 61(1) 4 Dimensional analysis 62(7) 4.1 Buckinghams theorem 62(1) 4.2 A simple example 63(1) 4.3 A less simple case 64(1) 4.4 Turbulent boundary layer 65(2) 4.5 Barenblatts second kind self-similarity 67(2) 5 Modern approaches in turbulence 69(10) 5.1 The Lorenz model 69(2) 5.2 Statistical mechanics for turbulence 71(3) 5.3 Developed turbulence 74(5) References 79(2) 2 Hydrodynamic instabilities in open flows 81(214) P. Huerre M. Rossi 1 Introduction 81(13) 1.1 An open flow example: the pipe flow experiment of Reynolds 84(6) 1.2 A closed flow example: Taylor-Couette flow between rotating cylinders 90(4) 2 Phenomenology of open flows 94(22) 2.1 The mixing layer as a prototype of noise amplifier 96(8) 2.2 The wake behind a bluff body as a prototype of hydrodynamic oscillator 104(7) 2.3 Plane channel flow as a prototype of a viscous instability 111(5) 3 Fundamental concepts 116(36) 3.1 Some formal definitions 117(2) 3.2 Linear instability concepts 119(26) 3.3 Nonlinear instability concepts 145(7) 4 Inviscid instabilities in parallel flows 152(17) 4.1 Squires transformation 154(1) 4.2 The two-dimensional stability problem: Rayleighs equation 155(3) 4.3 Rayleighs inflection point criterion 158(2) 4.4 FjXXXrtofts criterion 160(4) 4.5 Jump conditions at an interface: Application to the vortex sheet 164(5) 5 The spatial mixing layer 169(29) 5.1 Linear instability of parallel mixing layers 169(17) 5.2 Weakly non parallel WKBJ formulation 186(7) 5.3 Secondary instabilities 193(5) 6 The wake behind a bluff body 198(28) 6.1 Linear instability of locally parallel wakes 200(7) 6.2 Global instability concepts for spatially developing flows 207(13) 6.3 Phase dynamics of wake patterns 220(6) 7 Viscous instabilities in parallel flows 226(17) 7.1 Squires transformation 229(1) 7.2 The two-dimensional stability problem: the Orr-Sommerfeld equation 229(1) 7.3 A first look at the instability mechanism: the energy equation 230(2) 7.4 Heuristic analysis of the structure of two-dimensional Tollmien-Schlichting waves 232(11) 8 Plane channel flow 243(45) 8.1 Primary linear instability 243(5) 8.2 Weakly nonlinear analysis 248(8) 8.3 Finite-amplitude two-dimensional vortical states 256(10) 8.4 Universal elliptical instability 266(15) 8.5 Fundamental and subharmonic secondary instability routes 281(7) References 288(7) 3 Asymptotic techniques in nonlinear problems: some illustrative examples 295(92) V. Hakim Introduction 295(1) 1 Boundary layers and matched asymptotic expansions 295(14) 1.1 An elementary example of a boundary layer; inner and outer expansions 297(3) 1.2 Landau and Levich coating flow problem 300(9) 2 Multiscale analysis and envelope equations 309(16) 2.1 The period of the pendulum by the multiscale method 310(2) 2.2 The nonlinear Schrodinger equation as an envelope equation for small amplitude gravity waves in deep water 312(11) 2.3 Amplitude equation from a more general viewpoint 323(2) 3 Fronts and localized states 325(21) 3.1 Front between linearly stable states 325(13) 3.2 Invasion of an unstable state by a stable state 338(8) 4 Exponentially small effects and complex-plane boundary layers 346(36) 4.1 Introduction 346(3) 4.2 The example of the geometric model of interface motion 349(10) 4.3 The viscous finger puzzle 359(11) 4.4 Miscellaneous examples of exponential asymptotics in physical problems 370(12) References 382(5) 4 Pattern forming instabilities 387(106) S. Fauve 1 Introduction 387(7) 1.1 Example: the Faraday instability 387(4) 1.2 Analogy with phase transitions: amplitude equations 391(1) 1.3 Long-wavelength neutral modes: phase dynamics 392(1) 1.4 Localized nonlinear structures 393(1) 2 Nonlinear oscillators 394(16) 2.1 Van der Pol oscillator 395(6) 2.2 Parametric oscillators 401(4) 2.3 Frequency locking 405(5) 3 Nonlinear waves in dispersive media 410(14) 3.1 Evolution of a wave-packet 412(5) 3.2 The side-band or Benjamin-Feir instability 417(3) 3.3 Solitary waves 420(4) 4 Cellular instabilities, a canonical example: Rayleigh-Benard convection 424(14) 4.1 Rayleigh-Benard convection 424(5) 4.2 Linear stability analysis 429(3) 4.3 Nonlinear saturation of the critical modes 432(6) 5 Amplitude equations in dissipative systems 438(16) 5.1 Stationary instability 439(6) 5.2 Oscillatory instability 445(2) 5.3 Parametric instability 447(2) 5.4 Neutral modes at zero wavenumber. Systems with Galilean invariance 449(1) 5.5 Conserved order parameter 450(2) 5.6 Conservative systems and dispersive instabilities 452(2) 6 Secondary instabilities of cellular flows: Eckhaus and zigzag instabilities 454(19) 6.1 Broken symmetries and neutral modes 454(2) 6.2 Phase dynamics 456(1) 6.3 Eckhaus instability 457(12) 6.4 The zigzag instability 469(4) 7 Drift instabilities of cellular patterns 473(8) 7.1 Introduction 473(2) 7.2 A drift instability of stationary patterns 475(1) 7.3 The drift instability of a parametrically excited standing wave 476(2) 7.4 The drift bifurcation 478(1) 7.5 Oscillatory phase modulation of periodic patterns 479(2) 8 Nonlinear localized structures 481(8) 8.1 Different types of nonlinear localized structures 481(3) 8.2 Kink dynamics 484(2) 8.3 Localized structures in the vicinity of a subcritical bifurcation 486(3) References 489(4) 5 An introduction to the instability of flames, shocks, and detonations 493(182) G. Joulin P. Vidal 1 Introduction and overview 493(2) 2 Basic equations 495(4) 2.1 Conservation laws for reactive fluids 495(3) 2.2 Weak forms 498(1) 3 Subsonic versus supersonic traveling waves 499(4) 3.1 The (p-V) plane 499(1) 3.2 Various waves 500(1) 3.3 Shocks, detonations, and deflagrations 501(2) 4 Flames 503(89) 4.1 Phenomenology 503(4) 4.2 Minimal model and isobaric approximation 507(5) 4.3 The basic eigenvalue problem 512(15) 4.4 Jumps across the reaction layer 527(4) 4.5 Diffusive instabilities 531(9) 4.6 A conductive instability 540(6) 4.7 Hydrodynamic instability 546(22) 4.8 Body forces 568(9) 4.9 Hydrodynamic influence of boundaries 577(3) 4.10 Large-scale flow geometry 580(5) 4.11 Prospects 585(7) 5 Shock waves 592(35) 5.1 Phenomenology 592(2) 5.2 Shock formation 594(18) 5.3 Majda and Rosales model problem 612(2) 5.4 Dyakov-Kontorovichs instabilities 614(12) 5.5 Prospects 626(1) 6 Detonations 627(40) 6.1 Phenomenology 627(4) 6.2 Chapman-Jouguet model and sonicity condition 631(7) 6.3 Analogs of Dyakov-Kontorovichs instabilities 638(3) 6.4 Bruns model for autonomous diverging waves 641(8) 6.5 Zeldovich-Von Neumann-Doering model 649(8) 6.6 Chemistry-related instabilities 657(6) 6.7 Recent results and prospects 663(4) References 667(8) Index 675
