Physics of continuous matter: exotic and everyday phenomena in the macroscopic world

Benny Lautrup9780750307529, 0-7503-0752-8

Synopsis Continuum physics describes the physical world on a macroscopic scale. Despite the huge impact of quantum theory on physics in the 20th century, classical continuum physics is still an essential part of the physicists repertoire. Fields such as nonlinear science, geophysics, astrophysics, engineering and biophysics all require an understanding of the methodology of continuum physics, and the common principles underlying these very diverse topics at the macroscopic level. Offering a modern approach to this most classical of subjects, Physics of Continuous Matter is first and foremost an introduction to the basic concepts and phenomenology of continuous systems, and the derivations of the equations of continuum mechanics from Newtonian mechanics. Although many examples, particularly in earlier chapters, are taken from geophysics and astrophysics, the emphasis has been placed on generic methods and applications. Each chapter begins with a soft introduction, putting the discussion within an everyday context, and the level of difficulty then rises steadily, a pattern which is reflected throughout the text as a whole. The necessary mathematical tools are developed in parallel with the physics on a need-to-know basis, an approach which avoids lengthy mathematical preliminaries. Physics of Continuous Matter can be used as the main text for an early course on continuum physics or as the foundation for a series of related courses on more specialized topics. It will be an essential resource for physics students throughout the course of their degree and beyond.

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Table of contents :
Physics of Continuous Matter: Exotic and Everyday Phenomena in the Macroscopic World……Page 1
Contents……Page 3
Preface……Page 11
Molecular weight……Page 14
Avogadro’s number……Page 15
1.2 The continuum approximation……Page 16
Precision and continuity……Page 17
Material particles……Page 18
Global mechanical quantities……Page 19
Field equations……Page 20
Problems……Page 21
2.1 Reference frames……Page 23
2.3 Space……Page 24
Distance……Page 25
Basic algebraic rules……Page 26
Further definitions……Page 27
Handedness……Page 28
The Levi-Civita symbol……Page 29
Simple transformations……Page 30
Orthogonality and completeness of the new basis……Page 31
Vector quantities……Page 32
Gradient, divergence and curl……Page 33
Classification under reflections……Page 34
Problems……Page 35
3.1 Mass density……Page 39
3.2 Gravitational acceleration……Page 40
Total gravitational force and total moment of gravity……Page 41
3.3 Sources of gravity……Page 42
Field of a spherical body……Page 44
3.4 Gravitational potential……Page 45
Gravity as a gradient field……Page 46
The spherical case……Page 47
Escape velocity……Page 48
Problems……Page 49
Microscopic origin of pressure……Page 53
Incompressible sea……Page 54
4.2 Formal definition of pressure……Page 55
Same pressure in all directions?……Page 56
Global hydrostatic equilibrium equation……Page 57
The flat-Earth case……Page 58
Gauss’ theorem……Page 59
4.5 Barotropic fluid states……Page 60
Bulk modulus……Page 61
4.6 The homentropic atmosphere……Page 62
Isentropic processes in ideal gases……Page 63
The atmospheric temperature lapse rate……Page 64
Problems……Page 65
5.1 Archimedes’ principle……Page 67
Constant field of gravity……Page 68
Gas balloons……Page 69
Moments of weight and buoyancy……Page 70
Body floating on the surface……Page 71
5.4 Ship stability……Page 72
Displacement geometry……Page 73
The metacentre……Page 74
Stability condition……Page 75
Floating block……Page 76
Ship with liquid cargo……Page 77
Problems……Page 78
6.1 Gravitational flux……Page 80
Poisson’s equation……Page 81
Hydrostatic equilibrium……Page 82
Boundary conditions……Page 83
6.3 The homentropic star……Page 84
The stellar temperature lapse rate……Page 85
The Lane–Emden solutions……Page 86
Central values……Page 87
Assembly work……Page 88
Planet with constant density……Page 89
Problems……Page 90
7.1 Fluid interfaces in hydrostatic equilibrium……Page 92
Antigravity of rotation……Page 93
7.3 The Earth, the Moon and the tides……Page 94
The Moon……Page 95
The tides……Page 96
Quasi-static tidal cycles……Page 97
7.4 Shape of a rotating fluid planet……Page 98
Including the self-potential……Page 99
Total potential and strength of gravity……Page 100
Fast rotating planet……Page 101
Problems……Page 102
8.1 The Young–Laplace law for surface tension……Page 103
Pressure excess in a sphere……Page 104
Derivation of the Young–Laplace law……Page 105
8.2 Contact angle……Page 106
Capillary effect……Page 107
Geometry of planar curves……Page 108
The pendulum connection……Page 109
The capillary surface……Page 110
Pending droplets……Page 111
Problems……Page 113
Static and sliding friction……Page 115
External and internal stress……Page 116
Tensile strength……Page 117
The stress tensor……Page 118
Proof of Cauchy’s stress hypothesis……Page 119
Cauchy’s equation of equilibrium……Page 120
Boundary conditions……Page 121
The ambiguous stress tensor……Page 122
Problems……Page 123
10.1 Displacement……Page 125
Linear displacements……Page 126
10.2 Local deformation……Page 127
Small displacement gradients……Page 128
Cauchy’s strain tensor……Page 129
Changes in lengths and angles……Page 130
Change in volume and density……Page 131
Virtual displacement……Page 132
The Lagrange representation……Page 133
The Euler representation……Page 134
Euler versus Lagrange……Page 135
Problems……Page 136
Young’s modulus……Page 138
Poisson’s ratio……Page 139
11.2 Hooke’s law in isotropic materials……Page 140
Average pressure and bulk modulus……Page 141
Positivity constraints……Page 142
Uniform compression……Page 143
11.4 Energy of deformation……Page 144
Total energy in an external gravitational field……Page 145
Absolute minimum of energy in mechanical equilibrium……Page 146
Problems……Page 147
12.1 Equations of elastostatics……Page 149
Estimates……Page 150
Uniform settling……Page 151
Shear-free settling……Page 152
Centring the beam……Page 153
Total force……Page 154
Total moment……Page 155
Threshold for buckling……Page 156
12.4 Twisting a shaft……Page 157
Strain and stress……Page 158
Uniform radial displacement……Page 159
Equilibrium equation……Page 160
Strain and stress……Page 161
Solution for the pressurized clamped tube……Page 162
Unclamped tube……Page 163
Problems……Page 164
13.1 Relaxing towards equilibrium……Page 166
13.2 Discretization of space……Page 167
Finite difference operators with second-order errors……Page 168
13.3 Gravitational settling in two dimensions……Page 169
Shear-free solution……Page 170
Iteration cycle……Page 171
Results……Page 172
Problems……Page 173
Navier’s equation of motion……Page 174
Driving forces, dissipation and free waves……Page 175
Longitudinal and transverse waves……Page 176
Harmonic analysis……Page 177
Plane waves……Page 178
Snell’s law……Page 179
‘Supertransverse’ waves at an interface……Page 180
Reflection from a free surface……Page 181
Rayleigh waves……Page 183
Love waves……Page 184
Problems……Page 185
15.1 The velocity field……Page 188
Methods of flow visualization……Page 189
Relating the various flow lines……Page 190
Material rate of change of volume……Page 191
Leonardo’s law……Page 192
Global mass conservation……Page 193
Local material rate of change of density……Page 194
Material time derivative operator……Page 195
Newton’s Second Law for continuous matter……Page 196
Explosion field……Page 197
Evolution of the mass density……Page 198
Cosmic democracy……Page 199
The cosmic scale factor……Page 200
The cosmological constant……Page 201
Problems……Page 202
16.1 The Euler equation……Page 204
Wave equation……Page 205
Plane wave solution……Page 206
Bernoulli’s theorem for incompressible fluid……Page 207
Uses of Bernoulli’s theorem……Page 208
Torricelli’s Law……Page 209
The Pitot tube……Page 210
Shallow river with slowly changing bed……Page 211
16.4 Steady compressible flow……Page 212
Bernoulli’s theorem for barotropic fluids……Page 213
Stagnation point properties……Page 214
Ideal gas flow in duct with slowly varying cross section……Page 215
Asymptotically uniform flow……Page 217
Vortex lines……Page 218
Stokes’ theorem……Page 219
Kelvin’s circulation theorem……Page 220
Incompressible potential flow……Page 221
Pressure, lift and drag……Page 222
Potential and velocity field……Page 223
Pressure, lift and drag……Page 224
Effective mass in unsteady potential flow……Page 225
Problems……Page 226
17.1 Shear viscosity……Page 228
Molecular origin of viscosity in gases……Page 229
17.2 Velocity-driven planar flow……Page 230
Viscous friction……Page 231
Momentum diffusion……Page 232
Isotropic viscous stress……Page 233
The Navier–Stokes equations for incompressible fluid……Page 234
Viscous dissipation……Page 235
The Reynolds number……Page 236
Hydrodynamic similarity……Page 237
The Navier–Stokes equation……Page 239
The wave equation……Page 240
Damped plane wave……Page 241
Problems……Page 242
18.1 Steady, incompressible, viscous flow……Page 243
Specific solution……Page 244
Reynolds number……Page 245
Flow between inclined parallel plates……Page 246
Stability of open-surface flow……Page 247
General solution……Page 248
Reynolds number……Page 249
Pipe resistance……Page 250
Dissipation……Page 251
Laminar drain……Page 252
Friction factor……Page 253
Friction coefficient……Page 254
Rough pipe limit……Page 255
18.6 Circulating cylindrical flow……Page 256
General solution……Page 257
Stress, torque and dissipation……Page 258
Unloaded journal bearing……Page 259
Direction of secondary flow……Page 260
Rayleigh’s stability criterion……Page 261
Problems……Page 262
19.1 Steady incompressible creeping flow……Page 265
Estimates……Page 266
Stokes law for drag on a sphere……Page 267
Terminal velocity……Page 268
Limits to Stokes flow……Page 269
Drag coefficient……Page 270
Interpolation……Page 271
Terminal speed……Page 272
Streamlining……Page 273
Problems……Page 274
Velocity and acceleration in steady rotation……Page 277
Centrifugal and Coriolis forces……Page 278
Fictitious forces on Earth……Page 279
The Ekman number……Page 280
Isobaric flow and weather maps……Page 281
20.4 The Ekman layer……Page 282
Ekman layer solution……Page 283
Ekman upwelling and suction……Page 284
Rossby radius……Page 285
The Ekman layer valve……Page 286
20.6 Debunking an urban legend……Page 287
Problems……Page 288
21.1 Unsteady, incompressible flow……Page 289
Poisson equation for pressure……Page 290
Stability conditions……Page 291
Midpoint differences……Page 292
Double differences……Page 293
Discretized equations of motion……Page 294
Solving the discrete Poisson equation……Page 295
Initial data……Page 296
The grid……Page 297
Iteration cycle……Page 298
Results……Page 299
Problems……Page 300
22.1 Connected tubes……Page 303
Mechanical laws of balance……Page 304
Rate of change of a quantity……Page 305
Local and global rates of change……Page 306
Total force……Page 307
Launch of a small rocket……Page 308
22.6 Angular momentum balance……Page 309
Internal moment of force……Page 310
Spinning a rotating lawn sprinkler……Page 311
Internal rate of work……Page 312
Pulling the plug……Page 313
22.8 Mechanical energy balance……Page 314
Specific internal energy for non-viscous barotropic fluids……Page 315
Bernoulli’s theorem and energy balance……Page 316
Problems……Page 317
23.1 Reaction forces……Page 319
Reaction from draining cistern……Page 320
Nozzle puzzle……Page 321
Reaction force in steady flow……Page 322
Nozzle puzzle, continued……Page 323
Steady flow rotor physics……Page 324
Radial flow rotor……Page 325
Axial flow rotor……Page 326
Problems……Page 328
24.1 Basic physics of surface waves……Page 330
Collapse of a ‘waterberg’……Page 331
Gravity waves are nearly ideal……Page 332
24.2 Harmonic line waves……Page 333
Energy transport and group velocity……Page 334
Velocity potential……Page 335
Deep-water waves……Page 336
Harmonic line waves at finite depth……Page 337
Pressure jump across a nearly flat surface……Page 339
Deep-water capillary gravity waves……Page 340
Dispersion law……Page 342
The Rayleigh–Taylor instability……Page 343
The Kelvin–Helmholtz instability……Page 344
Total energy……Page 345
Rate of viscous dissipation……Page 346
Energy and amplitude attenuation coefficients……Page 347
The ‘canonical’ form of the spectrum……Page 348
The Pierson–Moskowitz empirical spectrum……Page 349
Mass flux……Page 350
Problems……Page 351
Stationary jump in planar horizontal flow……Page 354
Stationary circular jump……Page 356
Energy loss in the stationary jump……Page 357
Moving hydraulic jumps……Page 358
Thickness of a hydraulic jump……Page 359
25.2 Shocks in ideal gases……Page 360
Stationary planar normal shocks……Page 361
Oblique shock……Page 362
Front thickness……Page 363
Radius of the strong shock front……Page 364
Isentropic radial gas dynamics……Page 365
Strong self-similar shock……Page 366
Excess energy……Page 367
Problems……Page 368
26.1 Free cylindrical vortices……Page 369
Interpolating vortex……Page 370
Diffusive spin-down of vortex……Page 371
26.2 Ideal vortex dynamics……Page 372
The line vortex……Page 373
Complex notation……Page 374
The von Karman vortex street……Page 375
Vibrations driven by periodic vortex shedding……Page 377
Vortex equations……Page 378
Vortex with localized axial jet……Page 379
26.5 Advective spin-up of a vortex……Page 380
Inflow of angular momentum……Page 381
Choice of drain flow……Page 382
Estimating the depth of the central depression……Page 383
Numeric integration……Page 384
Problems……Page 385
27.1 Physics of lubrication……Page 388
Estimate of lift……Page 389
Form drag……Page 390
Solving the gap equations……Page 391
Flow reversal……Page 392
Pressure and lift……Page 393
Moment……Page 394
Narrow gap approximation……Page 395
Pressure and lift……Page 396
Problems……Page 397
Laminar boundary layer thickness……Page 399
Initial viscous growth……Page 400
Merging boundary layers……Page 401
Separation……Page 402
Analytic solution……Page 403
Thickness……Page 404
The Prandtl equations……Page 405
Upflow……Page 406
Blasius’ equation……Page 407
Laminar skin drag on a flat wing……Page 408
Friction coefficient……Page 409
Local drag and momentum balance……Page 410
Turbulent velocity profile and thickness……Page 411
The Falkner–Skan equation……Page 412
Upflow……Page 413
Accelerating and decelerating slip-flow……Page 414
Exact wall derivatives……Page 415
Exact integral relations……Page 416
28.8 Laminar boundary layer separation……Page 417
The wall-anchored model……Page 418
Momentum and energy balance……Page 420
Approximative solution……Page 421
Separation from cylinder……Page 422
Problems……Page 423
Control surfaces……Page 426
Take off, cruise and landing……Page 427
Total force……Page 428
29.3 Steady flight……Page 429
Steady climb……Page 430
Unpowered steady descent……Page 431
Wing and airfoil geometry……Page 432
Wing loading……Page 433
Velocity differences……Page 434
Circulation and lift……Page 435
The horseshoe vortex system……Page 436
Dependence on angle of attack……Page 437
29.5 Estimating drag……Page 438
Skin friction……Page 439
Form drag……Page 440
Induced drag……Page 441
Breguet’s range equation……Page 442
Momentum balance in a box……Page 443
Box at spatial infinity……Page 444
d’Alembert’s paradox: a gift to powered flight……Page 445
Lift and circulation……Page 446
29.7 Two-dimensional airfoil theory……Page 447
The Kutta condition……Page 448
The fundamental airfoil equation……Page 449
Thin airfoil approximation……Page 450
Oseen’s approximation……Page 451
Flow inside the wake……Page 452
Flow outside the wake……Page 453
Problems……Page 454
30.1 Energy balance……Page 457
Rate of heat transfer……Page 458
30.2 Heat equation for isotropic matter at rest……Page 459
Fourier’s equation……Page 460
Planar heat wave……Page 461
Steady heat production……Page 462
Incompressible inviscid fluid……Page 463
Compressible ideal gas……Page 464
General isotropic fluid……Page 465
Wind chill estimate……Page 466
Heat flow in the Blasius boundary layer……Page 467
Problems……Page 468
31.1 Convection……Page 470
Steady convection in open vertical slot heated on one side……Page 471
Entrance length for heat……Page 472
Thermal boundary layer……Page 473
Stability estimate for spherical blob of fluid……Page 475
The critical Rayleigh number……Page 476
Linearized dynamics of flow and heat……Page 477
Fourier transformation……Page 478
31.4 Rayleigh–B´enard convection……Page 479
General solution……Page 480
Two free boundaries……Page 481
Two solid boundaries……Page 482
Solid bottom and free top……Page 484
Problems……Page 485
32.1 Fully developed turbulence……Page 487
Scaling laws……Page 488
Energy spectrum……Page 489
Non-dimensional formulation……Page 491
Mean values……Page 492
Mean field equations……Page 493
Velocity defect……Page 494
32.4 Turbulence near a smooth solid wall……Page 495
Viscous linear sublayer……Page 496
Turbulent logarithmic sublayer……Page 497
Complete inner law of the wall……Page 498
Reynolds shear stress in the inner layer……Page 499
32.6 Outer layer structure……Page 500
32.7 Planar turbulent flows……Page 501
Pressure-driven planar flow……Page 502
Velocity-driven planar flow……Page 504
Wake function representation……Page 505
Velocity profile……Page 506
32.9 Turbulent boundary layer in uniform flow……Page 507
Momentum balance……Page 508
Plate drag coefficient……Page 510
Problems……Page 511
A.1 Dynamic equations……Page 512
A.2 Force and momentum……Page 513
A.3 Moment of force and angular momentum……Page 514
A.5 Internal and external forces……Page 515
Problems……Page 516
Curvilinear basis……Page 517
The Laplacian……Page 518
Resolution of the gradient……Page 519
The Laplacian……Page 520
C.1 Internal energy……Page 521
C.3 Entropy……Page 522
Isentropic processes……Page 523
C.4 Specific quantities……Page 524
Problems……Page 525
1 Continuous matter……Page 526
2 Space and time……Page 527
3 Gravity……Page 528
4 Fluids at rest……Page 531
5 Buoyancy……Page 533
6 Planets and stars……Page 536
8 Surface tension……Page 538
9 Stress……Page 540
10 Strain……Page 541
11 Elasticity……Page 544
12 Solids at rest……Page 545
13 Computational elastostatics……Page 547
14 Vibrations……Page 548
15 Fluids in motion……Page 549
16 Nearly ideal flow……Page 550
17 Viscosity……Page 553
18 Plates and pipes……Page 554
19 Creeping flow……Page 558
20 Rotating fluids……Page 560
21 Computational fluid dynamics……Page 561
22 Global laws of balance……Page 562
24 Surface waves……Page 564
25 Jumps and shocks……Page 566
26 Whirls and vortices……Page 567
27 Lubrication……Page 570
28 Boundary layers……Page 571
29 Subsonic flight……Page 572
31 Convection……Page 575
32 Turbulence……Page 576
C Thermodynamics of ideal gases……Page 577
List of literature……Page 578