Contents Preface xiii I Introduction 1 1 Continuous matter 3 1.1 Molecules................................ 4 1.2 The continuum approximation.................... 6 1.3 Newtonian mechanics......................... 10 1.4 Continuum physics........................... 11 Problems................................. 14 2 Space and time 15 2.1 Reference frames............................ 16 2.2 Time................................... 16 2.3 Space.................................. 17 2.4 Vector algebra............................. 20 2.5 Basis vectors.............................. 22 2.6 Index notation............................. 23 2.7 Cartesian coordinate transformations................. 24 2.8 Scalars, vectors, and tensors...................... 27 2.9 Scalar, vector, and tensor fields.................... 29 2.10 Pseudo and improper quantities.................... 30 Problems................................. 33 3 Gravity 37 3.1 Mass density.............................. 38 3.2 Gravitational acceleration....................... 40
iv CONTENTS 3.3 Sources of gravity............................ 43 3.4 Gravitational potential......................... 46 3.5 Potential energy............................ 49 Problems................................. 52 II Fluids at rest 55 4 Pressure 57 4.1 The concept of pressure........................ 58 4.2 Formal definition of pressure..................... 61 4.3 Hydrostatic equilibrium........................ 63 4.4 Equation of state............................ 67 4.5 Barotropic fluid states......................... 68 4.6 The homentropic atmosphere..................... 71 Problems................................. 75 5 Buoyancy 77 5.1 Archimedes principle......................... 78 5.2 The gentle art of ballooning...................... 80 5.3 Stability of floating bodies....................... 81 5.4 Ship stability.............................. 84 Problems................................. 90 6 Planets and stars 93 6.1 Gravitational flux............................ 94 6.2 Spherical bodies............................ 97 6.3 The homentropic star......................... 99 6.4 Gravitational energy.......................... 104 Problems................................. 107 7 Hydrostatic shapes 109 7.1 Fluid interfaces in hydrostatic equilibrium.............. 110 7.2 Shape of rotating fluids........................ 111 7.3 The Earth, the Moon and the tides.................. 113 7.4 Shape of a rotating fluid planet.................... 118 Problems................................. 123 8 Surface tension 125 8.1 Definition of surface tension...................... 126 8.2 Contact angle.............................. 130 8.3 Capillary effect at a vertical wall................... 131 8.4 Axially invariant shapes........................ 134 Problems................................. 138
CONTENTS v III Deformable solids 139 9 Stress 141 9.1 Friction................................. 142 9.2 The concept of stress.......................... 143 9.3 Nine components of stress....................... 145 9.4 Mechanical equilibrium......................... 148 9.5 Proof of symmetry of the stress tensor............... 150 Problems................................. 152 10 Strain 155 10.1 Displacement.............................. 156 10.2 Local deformation........................... 159 10.3 Geometrical meaning of the strain tensor.............. 162 10.4 Work and energy............................ 165 10.5 Finite deformations........................... 167 Problems................................. 171 11 Linear elasticity 173 11.1 Hooke s law............................... 174 11.2 Hooke s law in isotropic materials................... 176 11.3 Static uniform deformation...................... 181 11.4 Energy of deformation......................... 183 Problems................................. 187 12 Elastostatics 189 12.1 Equations of elastostatics....................... 190 12.2 Standing up to gravity......................... 192 12.3 Bending a beam............................ 195 12.4 Twisting a shaft............................ 198 12.5 Tube under pressure.......................... 200 Problems................................. 207 13 Computational elastostatics 209 13.1 Relaxing towards equilibrium..................... 210 13.2 Discretization of space......................... 211 13.3 Gravitational settling in two dimensions............... 214 Problems................................. 219 14 Elastic vibrations 221 14.1 Elastodynamics............................. 222 14.2 Refraction and reflection........................ 228 14.3 Surface waves.............................. 233 Problems................................. 238
vi CONTENTS IV Basic hydrodynamics 239 15 Fluids in motion 241 15.1 The velocity field............................ 242 15.2 Incompressible flow........................... 245 15.3 Mass conservation........................... 248 15.4 Moving along with the flow...................... 250 15.5 Continuum dynamics.......................... 252 15.6 Big Bang................................ 254 15.7 Newtonian cosmology......................... 256 Problems................................. 261 16 Nonviscous flow 263 16.1 The Euler equation........................... 264 16.2 Small-amplitude sound waves..................... 265 16.3 Steady incompressible flow....................... 267 16.4 Steady compressible flow........................ 275 16.5 Vorticity................................. 281 16.6 Circulation............................... 284 16.7 Potential flow.............................. 287 16.8 Potential flow for cylinder in cross-wind............... 288 16.9 Potential flow around a sphere in a stream.............. 290 16.10D Alembert s paradox......................... 292 Problems................................. 293 17 Viscosity 295 17.1 Shear viscosity............................. 296 17.2 Velocity-driven planar flow...................... 299 17.3 Incompressible Newtonian fluids................... 302 17.4 Classification of flows.......................... 306 17.5 Compressible Newtonian fluids.................... 310 17.6 Viscous attenuation of sound..................... 312 Problems................................. 315 18 Plates and pipes 317 18.1 Steady, incompressible, viscous flow.................. 318 18.2 Pressure-driven planar flow...................... 318 18.3 Gravity-driven planar flow....................... 321 18.4 Pipe flow................................ 323 18.5 Phenomenology of turbulent pipe flow................ 330 18.6 Circulating cylindrical flow...................... 335 18.7 Secondary flow and Taylor vortices.................. 340 Problems................................. 343
CONTENTS vii 19 Creeping flow 347 19.1 Steady incompressible creeping flow................. 348 19.2 Creeping flow around a solid ball................... 350 19.3 Beyond Stokes law........................... 353 19.4 Beyond spherical shape........................ 357 Problems................................. 359 20 Rotating fluids 363 20.1 Fictitious forces............................. 364 20.2 Flow in a rotating system....................... 367 20.3 Geostrophic flow............................ 368 20.4 The Ekman layer............................ 371 20.5 Steady bathtub vortex in rotating container............. 375 20.6 Debunking an urban legend...................... 378 Problems................................. 379 21 Computational fluid dynamics 381 21.1 Unsteady, incompressible flow..................... 382 21.2 Temporal discretization........................ 384 21.3 Spatial discretization.......................... 385 21.4 Channel entrance flow......................... 390 Problems................................. 397 V Special topics 399 22 Global laws of balance 401 22.1 Connected tubes............................ 402 22.2 Overview of the global laws...................... 403 22.3 The control volume........................... 404 22.4 Mass balance.............................. 406 22.5 Momentum balance........................... 407 22.6 Angular momentum balance...................... 410 22.7 Kinetic energy balance......................... 413 22.8 Mechanical energy balance....................... 417 22.9 Energy balance in elastic fluids.................... 418 Problems................................. 422 23 Reactions 425 23.1 Reaction forces............................. 426 23.2 Formal definition of reaction force.................. 429 23.3 Reaction moments........................... 432 Problems................................. 439
viii CONTENTS 24 Jumps and shocks 441 24.1 Hydraulic jumps............................ 442 24.2 Normal shocks in ideal gases...................... 449 24.3 Atmospheric blast wave........................ 453 Problems................................. 458 25 Small-amplitude surface waves 459 25.1 Basic physics of surface waves..................... 460 25.2 Harmonic line waves.......................... 464 25.3 Gravity waves.............................. 466 25.4 Capillary surface waves........................ 472 25.5 Internal waves.............................. 474 25.6 Energy and attenuation........................ 478 25.7 Statistics of wind-generated ocean waves............... 482 25.8 Global wave properties......................... 485 Problems................................. 487 26 Whirls and vortices 489 26.1 Free cylindrical vortices........................ 490 26.2 Ideal vortex dynamics......................... 493 26.3 Parallel line vortices.......................... 495 26.4 Steady vortex sustained by secondary flow.............. 499 26.5 Advective spin-up of a vortex..................... 502 26.6 Bathtub-like vortices.......................... 503 Problems................................. 508 27 Lubrication 511 27.1 Physics of lubrication......................... 512 27.2 Creeping flow in a long narrow gap.................. 515 27.3 Flat wing................................ 518 27.4 Loaded journal bearing......................... 521 Problems................................. 525 28 Boundary layers 527 28.1 Physics of boundary layers....................... 528 28.2 The Stokes layer............................ 532 28.3 Boundary layer theory......................... 535 28.4 The Blasius layer............................ 537 28.5 Turbulent boundary layer in uniform flow.............. 540 28.6 Boundary layers with varying slip-flow................ 545 28.7 Laminar boundary layer separation.................. 548 Problems................................. 555
CONTENTS ix 29 Subsonic flight 559 29.1 Aircraft controls............................ 560 29.2 Aerodynamic forces and moments................... 563 29.3 Steady flight.............................. 564 29.4 Estimating lift............................. 568 29.5 Estimating drag............................. 575 29.6 Lift, drag, and the trailing wake.................... 581 29.7 Two-dimensional airfoil theory.................... 587 29.8 The distant laminar wake....................... 592 Problems................................. 597 30 Heat 599 30.1 Energy balance............................. 600 30.2 Heat equation for isotropic matter at rest.............. 603 30.3 Heat equation for fluids in motion.................. 608 30.4 Advective cooling or heating...................... 611 Problems................................. 616 31 Convection 617 31.1 Convection............................... 618 31.2 Convective instability......................... 624 31.3 Linear stability analysis of convection................ 627 31.4 Rayleigh-Bénard convection...................... 631 Problems................................. 638 32 Turbulence 639 A Units and constants 641 B Newtonian particle mechanics 643 B.1 Dynamic equations........................... 644 B.2 Force and momentum......................... 644 B.3 Moment of force and angular momentum.............. 646 B.4 Power and kinetic energy....................... 647 B.5 Internal and external forces...................... 648 B.6 Hierarchies of particle interactions.................. 649 Problems................................. 650 C Curvilinear coordinates 651 C.1 Cylindrical coordinates......................... 651 C.2 Spherical coordinates.......................... 654
x CONTENTS D Thermodynamics of ideal gases 657 D.1 Internal energy............................. 657 D.2 Heat capacity.............................. 658 D.3 Entropy................................. 659 D.4 Specific quantities........................... 662 Problems................................. 663 Answers to problems 665 1 Continuous matter........................... 665 2 Space and time............................. 666 3 Gravity................................. 668 4 Fluids at rest.............................. 671 5 Buoyancy................................ 673 6 Planets and stars............................ 677 7 Hydrostatic shapes........................... 679 8 Surface tension............................. 680 9 Stress.................................. 681 10 Strain.................................. 682 11 Elasticity................................ 685 12 Solids at rest.............................. 686 13 Numeric elastostatics.......................... 688 14 Vibrations................................ 688 15 Matter in motion............................ 690 16 Nearly ideal flow............................ 692 17 Viscosity................................. 694 18 Plates and tubes............................ 696 19 Creeping flow.............................. 699 20 Rotating fluids............................. 701 21 Computational fluid dynamics..................... 702 22 Laws of balance............................. 703 23 Reaction forces and moments..................... 706 24 Jumps and shocks........................... 706 25 Surface waves.............................. 707 26 Whirls and vortices........................... 710 27 Lubrication............................... 714 28 Boundary layers............................ 715 29 Subsonic flight............................. 716 30 Heat................................... 719 31 Convection............................... 720 32 Turbulence............................... 721 A Units and constants.......................... 721 B Newtonian particle mechanics..................... 721 C Curvilinear coordinates........................ 721 D Thermodynamics of ideal gases.................... 721
xi Bibliography 723 Index 729
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