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List of Biographies

Name Figure Nr. Page Nr. Archimedes Fig. 2.19 37 Grigory Isaakovich Barenblatt Fig. 10.20 616 Henri-Émile Bazin Fig. 10.10 595 Fig. 3.22 93 Friedrich Wilhelm Bessel Fig. 8.13 452 Paul Richard Heinrich Blasius Fig. 9.19 516 Jean-Charles de Borda Fig. 3.40 122 Jacques Antoine Charles Bresse Fig. 3.49 134 Baron Augustin-Louis Cauchy Fig. 3.63 150 Elwin Bruno Christoffel Fig. 6.35 317 Benoit Paul Émile Clapeyron Fig. 7.30 413 Gaspard-Gustave de Coriolis Fig. 8.10 443 Maurice Marie Alfred Couette Fig. 7.12 380 Henry Philibert Gaspard Darcy Fig. 10.9 594 Paul Adrein Maurice Dirac Fig. 5.5 212 Johann Peter Gustav Lejeune Dirichlet Fig. 8.2 428 Vagn Walfrid Ekman Fig. 8.8 441 Hans Ertel Fig. 4.14 189 Fig. 3.19 88 Ferdinand Georg Frobenius Fig. 7.31 417 William Froude Fig. 7.25 399 Johann Carl Friedrich Gauss Fig. 5.1 200 George Green Fig. 5.2 201 Gotthilf Heinrich Ludwig Hagen Fig. 7.8 362 Oliver Heaviside Fig. 8.20 466 Hermann Ludwig Ferdinand von Helmholtz Fig. 4.10 176 Karl Hiemenz Fig. 9.2 491 Carl Gustav Jacob Jacobi Fig. 5.26 251 Nikolai Yergorowich Joukowski Fig. 3.56 142

© Springer International Publishing Switzerland 2016 621 K. Hutter and Y. Wang, Fluid and Thermodynamics, Advances in Geophysical and Environmental and Mathematics, DOI 10.1007/978-3-319-33633-6 622 List of Biographies

Name Figure Nr. Page Nr. Theodore von Kármán Fig. 9.15 509 Gustav Robert Kirchhoff Fig. 3.42 124 Martin Wilhelm Kutta Fig. 3.54 140 Josph-Louis Lagrange Fig. 5.23 246 Pierre-Simon Laplace Fig. 5.12 221 Gottfried Wilhelm Leibniz Fig. 3.12 74 Louis Melville Milne-Thomson Fig. 6.23 304 Richard von Mises Fig. 3.43 125 Claude-Louis Navier Fig. 7.3 357 Franz Ernst Neumann Fig. 5.13 222 Carl Gottfried Neumann Fig. 5.14 223 Fig. 3.17 84 Johann Nikuradse Fig. 10.13 600 Blaise Pascal Fig. 2.9 28 George W. Platzman Fig. 8.16 459 Karl Pohlhausen Fig. 9.29 545 Jean Léonard Poiseuille Fig. 7.7 361 Siméon Denis Poisson Fig. 8.6 435 Joseph Proudman Fig. 4.13 187 Markus Reiner Fig. 7.2 355 Georg Friedrich Bernhard Riemann Fig. 5.27 252 Carl David Tolmé Runge Fig. 3.55 141 Karl Hermann Amandus Schwarz Fig. 6.34 316 George Gabriel Stokes Fig. 7.4 358 Evangelista Torricelli Fig. 3.30 107 Erich Immanuel Trefftz Fig. 6.20 300 Name Index

A Bohlen, 488, 545, 549 Abel, 223 Boltzmann, 407 Abraham, 428 Bonham-Carter, 458 Abramowitz, 330, 451, 453 Borda, 59, 121–123, 172, 173, 311, 315 Aczel, 442 Boussinesq, 432, 481, 580, 603, 604, 619 Ahlfors, 273 Bowden, 456, 457 Aksel, 333 Bresse, 134 Allen, 619 Brown, 363 Appleton, 466 Buat, 619 Archimedes, 36–39, 51, 54 Buat-Nançay, 592 Athenaeus, 37 Buckingham, 614

B C Baral, 419 Caratheodory, 273 Barenblatt, 615–619 Carnot, 3, 128–131, 135, 311, 612 Bartholdy, 428 Catherine, 88 Batchelor, 364, 425, 468 Cauchy, 16, 58, 149–152, 160, 161, 179, 252, Bateman, 425 256, 278, 351, 357, 398 Bazin, 580, 592, 595 Chorin, 619 Becker, 7, 438 Christoffel, 316, 317, 320–323, 325, 327, 330, Belcher, 561 331 Benton, 511, 512 Chugunov, 419 Berker, 425 Clapeyron, 413, 414 Bernoulli, 2, 59, 79, 88, 92–95, 97–103, 105, Clausius, 3, 414 106, 109–113, 115, 116, 122–125, 130, Clebsch, 223 131, 133, 135, 137, 142, 154, 174, 175, Cochran, 511–513 215, 237, 239–241, 261, 262, 284, 288, Colebrook, 601 309, 312, 488, 495, 500, 550, 587, Colins, 558, 561 609–611 Cooper, 455–457 Bessel, 451, 452 Coppel, 503 Bingham, 8–10, 355 Coriolis, 15, 49, 160, 167, 169, 185, 442, 443, Bjerknes, 441 456 Blasius, 288–290, 295, 487, 506, 516–519, Cosserat, 161 522, 523, 533, 535–537, 542, 543, 550, Couette, 379–381, 383, 391, 579 558–560, 568, 569, 571–573, 592, 598, Csanady, 456–458 599, 601, 619

© Springer International Publishing Switzerland 2016 623 K. Hutter and Y. Wang, Fluid and Thermodynamics, Advances in Geophysical and Environmental Mechanics and Mathematics, DOI 10.1007/978-3-319-33633-6 624 Name Index

D Glen, 8, 370 d’Alembert, 24, 47, 221, 295 Goldstein, 558 Darcy, 579, 592, 594 Goy, 369 De Prony, 592 Graham, 369 De Waele, 8, 369 Green, 160, 179, 199, 201, 206, 207, 228, 230, Dennis, 558, 561 245, 364 Descartes, 88 Greve, 406 Dijkstra, 514 Dirac, 211, 212 Dirichlet, 225, 251, 252, 317, 427–429, 618 H Dombroklonskiy, 457 Hagen, 360, 362, 429, 480, 579, 583, 584, Dorsey, 362 586–589, 591, 593, 598 Drazin, 425, 432, 433, 480 Hagenbach-Bischoff, 361 Driscoll, 273 Hager, 69, 132, 288, 399, 516 Dryden, 425 Halley, 247 Dulong, 443 Hartree, 502–504, 506 Heaps, 449, 454–458 Heaviside, 465, 466 E Helmholtz, 175–177, 185, 188, 190, 250 Euler, 175 Henrici, 273 Einstein, 363, 364 Hensel, 428 Ekman, 440, 441, 445–450, 453–458, 480 Hertz, 466 Ellison, 453 Herwig, 487 Engler, 619 Hess, 67 Ertel, 188, 189 Hidaka, 458 Euler, 2, 5, 17, 49, 62, 64, 86–89, 91–93, 95, Hide, 188 156, 157, 162, 165, 185, 247, 273, 353, Hiemenz, 490, 491, 496, 497, 561 366, 446, 453 Hieron II, 37 Eytelwein, 592, 619 Hilbert, 300 Hobbes, 107 Hoff, 7 F Holstein, 488, 545, 549 Falkner, 490, 502–504, 517, 523 Homann, 496, 497 Fermat, 150 Hooke, 6, 8, 85 Finlayson, 432 Hospital, 550 Fitzgerald, 466 Howarth, 465, 467, 559–561 Fjeldstad, 457, 458 Huber, 579 Foristall, 458 Hunt, 124 Foucault, 443 Hussaini, 503, 523 Fourier, 428, 440, 463, 482 Hutter, 8, 9, 11, 12, 159, 179, 411, 438, 458 Frauenfelder, 579 Frederick, 247 Friedrich, 88 I Frobenius, 416, 417 Isenberg, 586 Froude, 132, 398, 399

J G Jacobi, 222, 250–252, 330, 476 Galilei, 107, 357, 587 Jefferey, 364 Gauss, 24, 38, 78, 91, 117, 119, 152, 199–201, Jöhnk, 179 203, 204, 206, 228, 265, 267 Johnson, 419, 559, 560 Gedney, 458 Jones, 456, 457 Gersten, 487, 488, 549, 553 Joukowski, 58, 140, 142, 144, 145, 263, 289, Gittler, 488 291, 292, 295, 297–300 Name Index 625

K N Katz, 188 Nansen, 441 Kelvin, 3, 165, 166, 172, 174, 231, 407, 466 Navier, 124, 356–358, 365, 366, 371, 425, 443, Kepler, 84 459, 465, 474, 487, 505, 529, 581, 583, Kirchhoff, 123, 124, 200, 311, 314 589 Kirchner, 419 Neumann, 222, 223, 225, 229, 251 Kneller, 84 Newton, 2, 6, 8, 52, 54, 74, 84, 85, 87–89, 109, Knopp, 273 117, 146, 351, 356, 359, 360, 369, 371, Knudt, 568 378, 379, 382, 388, 389, 392, 398, 416, Koebe, 300 419, 425, 507, 508, 527, 580, 583, 587, Kolmogorov, 616 592, 604 Kronecker, 141 Nigam, 512 Kummer, 141, 316 Nikuradse, 592, 599–601, 605–607, 617, 619 Kutta, 58, 140–142, 144, 145, 263, 289, 291, Noll, 179 292, 295, 297–300 Norton, 7, 370 Npoleon, 247

L O Lacomb, 458 Olver, 273, 291, 292 Lagerstrom, 425 Ostwald, 396 Lagrange, 85, 246 Lai, 457 Lakin, 503, 523 P Lamb, 187 Pannel, 598, 599 Lanchaster, 145 Pascal, 27, 28 Landau, 468 Pearce, 455–457 Laplace, 16, 207, 221, 229, 232, 365, 427, 481, Peter, 88 533 Petrow, 396 Laurent, 289 Piau, 380 Lehmann, 74 Piola, 124 Leibniz, 74, 75, 80, 81, 85, 88, 93, 191, 410 Platzman, 458, 459 Lick, 458 Pohlhausen, 545, 546 Love, 124 Poiseuille, 360, 362, 378, 389, 429, 480, 579, 583, 584, 586–589, 591, 593, 598, 601, 602, 618 Poisson, 199, 206, 209, 216, 224, 232, 425, M 427, 429, 434, 435, 481, 618 Madsen, 450, 452, 454, 455 Prandtl, 103, 104, 145, 288, 300, 429, 430, Mallock, 381 450, 456–458, 487, 497, 529, 546, 580, Mariotte, 108, 109 585, 586, 592, 599, 601, 603, 604, 606, Maupertuis, 247 607, 615, 619 Maxwell, 201, 466 Prien, 69 Mayer, 3 Prony, 619 McLaurin, 532 Proudman, 187, 433, 559, 560 McLeod, 514 Pythagoras, 297 McMeeking, 419 Mendelsohn, 428 Michel, 396 R Miclav˘ci˘c, 512, 514 Rao, 457 Milne-Thomson, 303, 304, 312, 314, 315 Rayleigh, 465, 467, 568, 569, 573 Möller, 173 Reiner, 7, 8, 354–356, 369 Moody, 601, 602 Reynolds, 58, 145, 146, 152, 358, 396, 456, Murnaghan, 425 473, 475, 477–480, 497, 505, 534, 562, 626 Name Index

579, 580, 589, 598, 599, 601, 605, 606, Toupin, 188 610, 618 Treder, 188 Ricatti, 5, 93 Trefethen, 273 Riemann, 150, 252, 256, 277–279, 287, 291, Trefftz, 298–302 316, 317, 322 Trösch, 8, 9, 11, 12 Riley, 425, 432, 433, 480, 558–561 Truesdell, 179, 188, 425 Rivlin, 8, 355 Riwlin, 8, 354–356 Robbin, 559 U Robins, 560, 561 Unger, 438 Rosenhead, 558 Rowell, 432 Runge, 140, 141, 300 V Venn, 88 Vitruvius, 37 S Von Humboldt, 251 Saint-Venant, 425, 580 Von Kármán, 69, 298, 299, 301, 302, 450, 491, Schlichting, 425, 488, 549, 553 509–514, 545, 546, 592, 601, 605, 607, Schneider, 488 619 Schröder, 188 Von Mises, 124, 125, 300 Schukowski, 142 Schwarz, 316, 320–323, 325, 327, 330, 331, 436 W Segner, 153–156, 162 Walz, 551, 552 Serrin, 514 Wang, 425, 512, 514 Shanahan, 273, 330, 332 Watson, 330 Shaw, 273 Weatstone, 466 Sherman, 468 Weber, 200 Skan, 490, 502–504, 517, 523 Weierstrass, 141, 316 Sommerfeld, 396 Weisbach, 594 Spurk, 333 Squire, 468 Weiss, 222 Stanton, 598, 599 Welander, 458 Stegun, 330, 451, 453 Weyl, 503, 523 Stein, 74 Wheeler, 434 Steiner, 34, 252 White, 601 Stokes, 6, 138, 139, 163, 169, 171, 172, 199, Whitham, 425 205, 218, 227, 228, 267, 269, 356–358, Whittaker, 330 360, 365, 366, 371, 387, 419, 425, 459, Wilshinsky, 419 461, 465, 474, 487, 505, 529, 581, 583, Witten, 457, 458 589 Svensson, 455 Y Young, 5 T Taylor, 53, 54, 101, 187, 211, 257, 300, 303, 364, 381, 383, 416, 448, 616 Z Thirot, 512 Zandbergen, 514 Thomas, 457, 458 Zhukowsky, 142 Thomson, 201 Zill, 273, 330, 332 Torricelli, 28, 106–109, 587, 589 Zima, 396 Subject Index

A of , 113, 129, 407 , 24, 47, 49 of hydrostatic pressure, 400, 444 absolute, 48 of internal energy, 414 centripetal, 49, 51 of , 564 Corilis, 48 of linear , 59, 83, 147, 152, 168, Euler, 49 363 guiding, 48 of , 31, 59, 71, 73, 75–79, 92, 125, of a particle, 61 135, 176, 186, 190, 259, 352, 353, 364, relative, 48 410, 426, 488, 583 Acceleration distance, 587 of mass (or volume), 212, 255, 311, 314, Accumulation-ablation (rate) function, 410, 591 412, 415 of mass for a density-preserving fluid, 499 Activation energy, 407 of mass, global form, 538 Advection-diffusion-reaction equation, 414 of mechanical energy, 59, 92, 113, 114 Aerostatics, 17 of of momentum, 146, 351 Archimedean of momentum, 85, 95, 117, 123, 126, 127, buoyancy , 119 130–132, 146, 149, 351, 373, 400, 488, buoyancy force formula, 38, 39 542, 548, 564, 581 principle, 36, 51, 54 of momentum, boundary layer, 543, 569 Arrhenius law, 407 of momentum, global form, 90, 91, Aspect ratio, 40, 41 119–121, 350, 538 Asymptotic expansion, 487 of momentum, global formulation, 89 matched, 487, 499, 529, 534 of momentum, local form, 351 Atmosphere of potential vorticity, 190, 194 ipolytrope, 52 of static moment, 33 isothermal, 52 of vertical momentum, 444 troposphere, 53 Barometric height formula, 54 Autonomous, 66 Barometric pressure formula, 101 non-, 66 Barotropic Axisymmetric laminar jet, 467 density-preserving fluid, 191 fluid, 26, 97, 175, 177, 185, 191, 250, 353 gas, 97 B ideal fluid, 165, 166, 170, 172, 174, 177, Balance (law) 186, 188, 190 local conservation form, 352 inviscid fluid, 168, 174 of , 59, 88, 146, 148, potential vorticity, 192 151–154, 156, 161, 162 pressure, 26 of angular momentum, global form, 152, Bernoulli 161

© Springer International Publishing Switzerland 2016 627 K. Hutter and Y. Wang, Fluid and Thermodynamics, Advances in Geophysical and Environmental Mechanics and Mathematics, DOI 10.1007/978-3-319-33633-6 628 Subject Index

constant, 135 407, 426, 433, 439, 448, 450, 461, 467, equation, 59, 79, 92–95, 97–103, 105, 106, 475–477, 479, 481, 489, 499, 502, 509, 109–113, 115, 116, 122, 123, 125, 130, 513, 514, 518, 519, 522, 530, 531, 533, 133, 137, 142, 154, 215, 237, 239–241, 534, 537, 542, 545, 555, 556, 562, 563, 261, 262, 284, 288, 309, 312, 488, 495, 567, 582, 584, 590, 591 500, 587, 608–610 basal, 403 equation for a barotropic fluid, 97 basal surface, 388 equation, extended, 611 Dirichlet, 618 extended equation, 609 free surface, 388 number, 88 Neumann, 229 pressure difference, 131 no-slip, 416, 474, 476 surface, 174, 175 outer, 499 type equation, 130 Boundary layer, 485, 615 Bernoulli-Hospital rule, 550 (2D) near a stagnation point, 488 Bernoulli-Navier beam theory, 124 (3D) near a stagnation point, 494 Bessel’s differential equation, 451 along a flat plate, 539 Bifurcating streamline, 259, 260, 262 along a semit-infinite plate, unsteady, 572 Bingham along sidewall of wedge, 499 fluid, 9 around wedge, 497 fluid behavior, 8 assumption, 523, 524 medium, 10 atmospheric, 453 type material, 9 thickness, 488, 492–494, 504, Blasius 505, 540, 541, 569 boundary layer, 506, 516, 523, 533, 536, due to an oscillating body, 561 537, 542, 543, 550, 568, 571, 573 due to impulsive start of a , 554 boundary layer problem, 487 equation, 485, 497, 499, 525, 538 boundary layer solution, 573 equation, near-wall, 569 boundary value problem, 517, 518, 536 equation, two-dimensional, 498, 554, 561 differential equation, 517, 522 equation, x-momentum, 567 equation, 523, 537 momentum thickness, 488, 540, 541, 569 flow, 487, 502, 569 near free surface, 417–419 formula, 289, 290, 295 non-stationary, 553 problem, 523 non-stationary plate, 568 profile, 517, 522 thickness, 461, 465, 478, 505, 506, 511, solution, 519, 523, 533, 535, 537, 568 512, 517, 540–542, 562, 568, 572, 573, Body 614 elastic, 3 turbulent, 450, 615, 619 plastic, 3 Boussinesq formula, 604 rigid, 3 Brunt-Väisälä frequency, 55 viscous, 3 Buoyancy force, 38–43, 54 Body force, 19, 22, 23, 52, 351, 459 Buoyancy frequency, 55 conservative, 165, 166, 169, 185 free of vorticity, 25 irrotational, 25 C Potential, 25 Carnot potential, 165, 185 abrupt energy loss, 128, 129 specific, 19, 22–24, 47, 85, 96, 98, 119, cycle, 413 161, 371, 387, 398, 583 pressure loss, 130, 612 virtual, 236 principle, 413 Boltzmann’s constant, 407 shock loss, 128 Borda exit orifice, 120, 121, 123 Cauchy’s integral formula, 336 Boundary condition, 206, 208, 209, 221, 231, Cauchy’s integral theorem, 344 244, 372, 376, 377, 382, 383, 388, 391, Cauchy’s residue theorem, 344 Subject Index 629

Cauchy’s theorem, 334, 335 Conservation (law) Cauchy-Green deformation tensor, 179, 182 of angular momentum, 159, 161 Cauchy-Riemann of circulation, 280 differential equation, 252, 256, 271, 275, of energy, 176 276 of mass, 70, 73, 74, 77, 79, 255 equation, 278, 334 of moment of momentum, 161, 381 Center of , 37–41, 44, 46, 51, 85 of momentum, 471 Center of pressure, 33 of potential vorticity, 188, 190 Characteristic of source, 280 slope, 618 of volume for constant density, 255 Characteristic equation, 180, 183 Conservative quantity, 64 Characteristics, 155, 368, 572, 573 Constitutive Characteristic straight, 570 assumption, 390 Circulation, 137–139, 143–145, 163, 165, 167, behavior, 414 169–171, 205, 228, 229, 249, 254, 263, equation, 398 269, 288, 292, 297 model, 418 conservation (law), 280 modeling, 124 due to , 169 parameterization, 400 due to pressure, 169 postulate, 407 due to viscous stress, 169 relation, 5, 353, 354, 356, 366, 374, 398 flow, 286 Continuity equation, 76, 78, 80, 106, 129, 133, of vortex tube, 172 388, 399, 400, 403, 459, 468, 583, 610 Closure assumption, 604 Convex Closure condition area, 264 k − ε, 455 body, 264–266 turbulent, 580 part, 266 Closure relation two-dimensional cylindrical body, 257 for shear stress, 444 Convexity, 264 phenomenological, 580 Coriolis Coefficient of resistance, 592, 598, 599, 601, acceleration, 48 602, 607 effect, 456 turbulent, 604 force, 169, 185, 441 Colebrook-White formula, 601 parameter, first, 167, 442 Complex function, 273, 318 parameter, second, 167, 442 Complex function, 318, 320 Corner flow, 503 Complex , 278, 279, 283, 285, 289, 293 Cosserate material, 161 plane, 279 Couette potential, 252, 331 flow, 379, 381, 383, 391 Compressible fluid, 353 viscometer, 379 Conduit Coulomb , 386 circular, 430 Creep cylindrical, 427 flow, 397 triangular, 434 flow of a pseudoplastic fluid, 397 with arbitrary but constant cross-section, fluid, 414 427, 429 of isotropic polycrystalline ice, 371 Configuration, 60 of polycrystalline material, 370 current, 61 primary, 5 of the body at , 60 response function, 368, 405, 414 present, 60, 61, 181, 183 secondary, 5 reference, 60–62, 183 stationary, 5 Conformal mapping, 142, 276, 277, 279–282, steady state, 5 286, 288, 290, 296, 302, 310, 316, 319, tertiary, 5 322, 327, 328, 330–333, 429 transient, 5 630 Subject Index

Creep curve, 4, 5, 368, 369, 379, 384 induced flow, 212 concave, 369 moment of closed vortex filament, 220 convex, 369 plane flow, 257 of polycrystalline ice, 370 singularity, 303 Creep law, 370, 418 Dirichlet for dilatant fluid, 7 boundary value problem, 427, 429 for pseudoplastic fluid, 7 problem, 225 Critical water depth, 133 Dissipation Current rate of turbulent kinetic energy, 455 drift, 440, 453, 454 specific, 564 surface, 446, 448, 449, 453 turbulent, 580 wind drift, 448 Dissipative wind-induced, 449 force, 564 power, 564 Distortion (rate) tensor, 360 D Divergence theorem, 78, 171, 203, 233, 436 D’Alembert’s Dome, 402, 405, 406, 412, 415, 417–419 inertial force, 24 point, 405 paradox, 295 Doublet, 212, 213, 258 principle, 24, 47 field, 213 Deflection angle, 446, 448, 449, 453, 455, 458 plane, 257, 259 Density source, 258 of pure water, 11 Drag coefficient, 145, 146, 386, 404, 412, 419, of salt water, 12 506, 507, 544 Density preserving, 10–12, 17 effective, 386 fluid, 26, 27, 32, 36, 38, 50, 51, 54, 59, 75, Drift velocity, 568 76, 78, 80, 94, 98–100, 104, 109, 116, , 2 119, 125, 177, 178, 186, 190, 192, 273, 277, 284, 307, 331, 353, 360, 364–366, 371, 379, 381, 384, 387, 425 E material, 77, 353, 388 Eigenvalue, 179–181, 183, 354 Derivative, 61 equation, 180, 184 advective, 63 problem, 183 convective, 63 Ekman local, 62, 63 -type solution, 458 material, 61, 62, 426 current meter, 442 material time, 62 depth, 445, 446, 448 substantive, 61, 62 flow, 480 temporal, 61 layer, 449, 450, 455, 456 Diffusivity, 83, 412 problem, 449, 450, 456–458 Dilatant fluid, 7, 10, 366, 368–370, 373, 388, solution, 453, 456 389 spiral, 446–448, 454 Dimensional analysis, 592, 598 spiral above the bottom surface, 450 Dimensional homogeneity, 597 spiral below the water surface, 444 Dimensional matrix, 614 theory, 440, 441, 454, 455, 458 Dimensionally homogeneous, 362, 593 water bottle, 442 Dipole, 213 Ekman curent field, 213 finite depth, 448 flow, 214, 243, 286 infinite depth, 445 flow bounded in sphere, 214 non-constant vertical eddy viscosity, 450 flow field, 213 Elliptic integral of the first kind flow potential, 260 complete, 329 flow, two-dimensional, 257 standard, 329 Subject Index 631

Energy height, 610 definition, 69 Energy level, 610 Prandtl, 104 Energy loss, 129, 130, 132, 587 Fluidity, 367, 370, 399, 412 Carnot’s abrupt, 128, 129 Force coefficient, 130 on a body of arbitrary geometry, 241 in hydraulic jump, 134 on a body, -induced, 238 Entrainment, 527 on a cylinder, 262 rate, 80 on a sphere, 239 Entrance distance, 587 on a sphere with variable radius, 236 Entrance length, 587 Form parameter, 547 Entrance loss, 611 Fourier Entropy, 3 decomposition, 440 Equi-potential line, 253, 258, 260, 488 integral, 463 Equilibrium configuration, 38, 43, 45–47 Frame admissible, 45 inertial, 15, 98, 167, 169 indifferent, 38 non-inertial, 15, 168, 236, 239, 241, 440 stable, 38 Frictional (stress) tensor, 351, 444 unstable, 38 Frictionless fluid, 84 Euler Frobenius expansion, 416 constant, 453 Froude number, 398 equation, 86–89, 91, 92, 95, 175, 353, 366 Fundamental hydrostatic equation, 22, 24, 25, equation of a barotropic fluid, 185 47, 49, 51, 52 fluid, 165 formula, 273, 446 turbine equation, 156–158, 162 G Eulerian Galilei transformation, 71 description, 62, 64 Gas constant, 101 representation, 62 specific, 52 Gauss law, 24, 38, 78, 91, 117, 119, 152, 199, Evolution equation, 190, 411 203, 204, 206, 228, 264, 265, 267, 436, for thickness of fluid film, 82 437 Geodetic height, 610 Geothermal heat flow, 407, 408 F Green identity, 199, 206 Falkner-Skan first, 206, 228, 230 boundary value problem, 502, 503 second, 206, 207, 245 equation, 490, 502, 503, 517 Grounding line, 398 problem, 523 profile, 503, 504 Field theory, 2 H Floating behavior of beam, 43 Hagen-Poiseuille Floating body, 36, 38, 41, 42 equation, 361 Floating configuration of beam, 44 flow, 583, 584, 588, 591, 593, 598 Flow formula, 582, 587, 589 around cylinder, 260, 261 law, 362, 580, 586 around rigid circular kernel, 254 Hagen-Poiseuille flow, 427, 429, 480 around semi-infinite blunt body, 259 for elliptical cross-section, 429 around sphere, 213, 214 for general cross-section, 427, 434 around three-dimensional rigid body, 235 for rectangular cross-section, 432 Flow filament, 73, 75, 76, 95, 169 for triangular cross-section, 432 definition, 69 Harmonic plane, 94 field, 221 Flow out of a vessel, 586, 590 function, 222 Flow tube, 76, 103, 113–115, 135, 169 vector field, 221 632 Subject Index

velocity field, 209 approximation, 533 Harmonic function, 332 expansion, 546 Hartree profile, 503–506 region, 498, 531, 533 Heat solution, 531, 532, 534, 535, 537 flux vector, 407 Interaction force, 17 Heat conductivity, 407 attractive, 17 Helmholtz between two molecules, 17 equation, 177 cohesive, 17 theorem, 177 intermolecular, 18 Hodograph plane, 447 repelling, 17 Holomorphic function, 252, 253, 276–278, Internal energy, 407 289, 321 Inviscid/ideal fluid, 84, 85, 88, 95, 124, 144, Holstein-Bohlen procedure, 488, 545, 549 155, 163, 165, 169, 177, 238, 488, 493, Homogeneous prestress, 585 497, 498, 541, 587, 588 Hooke’s law, 6 barotropic, 168, 174 Hooke’s law of elasticity, 6 density-preserving, 253 Hookean material, 6, 8 past an infinitely long wedge, 499 Hydraulic Irrotational heaver, 32 acceleration field, 166 press, 28, 29 body force, 25 Hydraulic jump, 131–135 field, 139, 209 Hydraulic pressure, 28 flow, 139, 174, 227–229, 231, 235, 251 Hydraulically rough flow around a sphere, 213 flow regime, 619 flow field, 139, 166 pipe, 607, 613 flow, parallel, 215 regime, 614 motion, 231 Hydraulically smooth, 598, 601 part, 176 behavior, 619 potential flow, 553 pipe, 598, 599, 602, 604–607 vector field, 166, 221 turbulence, 601 velocity, 209, 210 Hydrodynamic lift, 142 velocity field, 226, 228 Hydrodynamic normal force, 142 Isobaric surface, 49, 50 Hydrodynamics, 60, 62, 142 Isochoric of ideal fluid, 143 motion, 10 of ideal liquid, 57 (volume preserving) transformation, 10 Hydrostatic buoyancy, 36 Isotropic compression, 357, 359 Isotropic expansion, 359 Hydrostatic pressure, 21, 22, 28, 401, 609 balance, 400 condition, 131 J contribution, 401 Jacobian equation, 444 determinant, 250 Hydrostatics, 17, 22, 28 elliptic function, 330, 476 in accelerated system, 47 in rotating system, 49 in translatorically accelerated system, 51 K Kelvin’s (circulation) theorem, 165, 166, 172, 174 I Kelvin’s energy theorem, 231 Ideal gas, 100 Kinematic (surface) equation, 82, 403, 410, Incompressible, 10 412 Inertial force, 24, 239, 241 Kinematic wave equation, 81, 82 virtual, 561 , 2 Inner Kinetics, 2 Subject Index 633

Kutta-Joukowski domain, 60 aerofoil, 142 particle, 60–62 condition, 144, 145, 292, 297 point, 18, 60, 73 criterion, 145 region, 60 formula, 141, 143, 263 surface, 81, 118 hypothesis, 143 volume, 19, 73, 89, 90, 92, 113, 117, 118, mapping, 290, 291, 295, 298–300 120 profile, 298, 299 volume element, 85 theorem, 289, 295 Material equation, 353, 356, 388, 399 transformation, 298 Material relation, 353 Materially constant, 64 Maximum-minimum property, 232 L Mean value theorem of potential theory, 232 Lagrangean Metacenter, 42, 43, 46 description, 62 Metacentric height, 42, 43 equation, 246, 247 Method Laminar flow, 579 function-theoretical, 273, 307 in a pipe of arbitrary cross-section, 583 of complex valued function theory, 124 in a pipe of elliptical cross-section, 584 of conformal mapping, 429 pipe, 580 of dimensional analysis, 598 Laplace of kinematics, 3 equation, 207, 229, 232, 253, 481, 584 of matched asymptotic expansions, 419, operator, 365, 427, 533, 584 487, 499, 529 Laurent expansion, 289 of singular perturbation, 487 Laurent series, 290, 338, 341–343, 345 of , 3 Laurent’s theorem, 338, 339 particle image velocimetry (PIV) , 67 Leibniz rule, 75, 81, 191, 410 particle tracking velocimetry (PTV), 67 Lift coefficient, 143, 293, 297 separation of variables, 72 Lift force, 290, 293, 297 Micromorphic material, 161 hydrodynamic, 142 Micropolar material, 161 Logarithmic Milne-Thomson singularity, 451, 454, 458, 472 (circle) theorem, 303, 312 spiral, 446, 447 method, 315 velocity deficit, 453 Moment of , 33, 34 Logarithmic law, 615 areal, 42 universal, 615, 618 polar, 35 Logarithmic velocity profile, 605, 614 Lubrication dynamic, 396 N of gears, 396 Navier-Stokes equations, 357, 358, 365, 366, 371, 425, 474, 487, 529, 581, 583, 589 M equations of a compressible fluid, 364 Manometer, 29–31, 102 equations of a density preserving fluid, 365 fluid, 103 fluid, 356, 459, 465, 505 liquid, 103 Neumann U-shaped, 29 boundary condition, 229 U-tube, 29–31 problem, 222, 225 Material Newton’s body, 91 fundamental law, 85, 87, 117, 146 coordinate, 62 gravity law, 52 derivative, 61–63 law, 87, 89, 398 description, 62 method of fluxions, 88 634 Subject Index

principle, 2 Plug flow, 389 second law, 54, 85, 88, 351, 398 Poiseuille flow, 389, 390, 601, 602 viscous law, 580 plane, 378 Newtonian Poisson equation, 199, 206, 209, 216, 224, 232, behavior, 369 429, 434, 584, 618 fluid, 6–8, 356, 359, 360, 369, 378, 379, Polar decomposition, 179–181 382, 389, 392, 416, 507, 508, 527, 583, theorem, 179, 180, 185 587, 604 Polar theory, 161 material behavior, 592 Potential, 310 No-slip condition, 583–585 of body force, 25, 185 Non-convex, 264 of flow around circle, 287 area, 264 of flow around sphere, 287 body, 266, 267 of parallel flow, 243 Non-Newtonian of parallel flow around circle, 304 fluid, 6, 8, 10 of plane flow, 303 fluid equation, 388 of two-dimensional dipole flow, 257 material behavior, 592 Potential field, 229 No-slip condition, 384, 388, 403, 404, 412, Potential flow, 229–232, 235, 236, 238, 415, 416, 474–476, 498, 514, 536, 560 254–256, 260, 263, 277, 282, 286–288, 290, 366, 488, 489, 494, 496, 534, 538, 541, 554, 558, 561, 567 O along a circular segment, 297 Oil spill trajectories, 455 around a circle, 287, 288, 292, 304 Orifice coefficient, 125 around a circular body, 277, 278 Oscillation-induced drift current, 566 around a circular segment, 298 Outer around a corner, 283 approximation, 533 around a front edge, 294 expansion, 546 around a sphere, 287 region, 498, 531, 533 around an arbitrary cylinder, 287, 288 solution, 530–532, 534–537 around an arbitrary two-dimensional body, Overlapping 287 condition, 535 around and past object, 331 region, 531, 532, 535, 537 down a step, 333 down or up a finite step, 333 P down or up a step, 334 Particle image velocimetry (PIV), 67, 405 from a bottom duct into a still fluid Particle tracking velocimetry (PTV), 67, 405 half-, 332 Pascal paradox, 27, 29 from a concentrated source at the channel Perturbation end, 318 expansion, 419, 530 from a point source, 286 expansion, regular, 419, 531, 556 from an underwater cantilever plate, 331 problem, singular, 511, 531 in a 2D rectangular and polygonal region, regular, 529 318 singular, 487 in a wedge, 283, 284 solution, 534 induced by a point source, 304 technique, singular, 533 onto a plate, 314 theory, singular, 419 onto a rigid immobile plate, 318 Phase , 461 out of a slit orifice, 307 Piezometric pressure, 99, 101–103, 119 over a circular segment, 295 Pipe flow, 577 over a continuous step, 318 laminar, 579 over a plane plate, 292 turbulent, 579 over abrupt step, 315 Plane film flow, 387–389 over an abruptt step, 318 Subject Index 635

over an underwater cantilever plate, 332 Pressure height, 610 parallel, 303 Pressure line, 610 through a narrowing width, 318 Pressure loss, 130, 587, 593, 598, 599, 612, through a periodic arrangement of slits, 311 613 through a rectangular and polygonal cross Carnot, 612 section, 318 coefficient, 609, 610 through a slit orifice, 307 coefficient, total, 611 through an exit opening, 315 curvature, 612 up a step, 333 due to curvetre, 613 with point singularities, 303 due to friction, 612, 613 Potential vortex, 164, 228, 254 due to sudden cross section enlargement, doublet, 258 612 flow, 229, 253, 256, 261 due to viscosity, 609 flow around a right circular kernel, 254 height, 610 Potential vorticity corollary, 190 in conduit, 611 Power law, 368–371, 388, 389, 407 in pipe flow, 608 creep response function, 414 normalized, 593 Glen, 8, 369, 407 term, 610 Norton, 370 Pressure-drag Power of working, 155, 156, 158, 565 approximation, 398, 412 available, 156 flow, 372, 375, 397, 398, 400, 405, 414 maximum possible, 156 flow with free surface, 3D, 397, 405 of fluid machine, 113, 116 mechanism, 375 of force, 246 theory, 406 of pump, 116, 158, 386 Principle of universality, 3 of the dissipation, 566 Process usaable, 156 reversible, 413 Prandtl Pseudoplastic creep law for ice, 406 boundary layer equation, 487, 529 Pseudoplastic fluid, 7, 366, 368, 369, 373, 388, 389, 397–399, 416 flow tube, 103, 104 Pseudoplastic shear thinning fluid, 10 formula, 601 formula for turbulent shear stress, 603 layer, 456–458 R membrane analogy, 585, 586 Rayleigh mixing length, 604 boundary layer, 573 soap film, 585 method, 571 soap film analogy, 429, 430 problem, 465, 467 stagnation tube, 104 solution, 568, 569 theory of turbulent boundary layer, 450 Reducible closed path, 226 tube, 104 Reference system, 89, 181, 239 viscous layer, 456 accelerated, 47 Prandtl-Colebrook formula, 601 non-inertial, 47 Prandtl-von Kármán turbulence theory, 607 Region Pressure, 20, 21 connected open, 226 barotropic, 26 n-fold connected, 226 fluid, 19, 21 simply connected, 226 hydrostatic, 21, 22 Representation Pressure distribution, 215, 488 Cartesian, 63 hydrostatic, 401 coordinate invariant, 62 in a density preserving fluid, 26 Eulerian, 62 in a pipe, 588, 590, 609 spatial, 62 in pressure-drag flow, 375 Residue theorem, 289, 343 on the surface of a sphere, 215 Reynolds 636 Subject Index

number, 145, 146, 358, 456, 473, 475, method, 72 477–480, 497, 505, 534, 562, 589, 598, solution, 439, 468 599, 601, 605, 606, 610, 618, 619 Shallow stress, 580 flow approximation, 409, 416–419 transport theorem, 91, 152 flow problem, 419 Reynolds number, 615, 619 ice approximation, 406, 407, 419 global, 614–616 water approximation, 168, 186, 191, 192, local, 614 194 Riemann Shallowness mapping theorem, 316, 322 approximation, 401 sheet, 279, 291 assumption, 412 theorem, 277, 287 Shear angle, 4, 5, 350 Rotating disk, 512 Shear deformation, 4 impulsively started, 512 Shear modulus, 5 of infinite extent, 507 effective, 5 of inite extent, 508 Young’s, 5 steady flow, 515 Shear stress, 614 Rotation matrix, 181 wall, 614 Roughness, 593 Shear stress velocity, 604 absolute, 602 Shear-strain relation, 4 length, 598 Shear thinning of a wall, 596 behavior, 366 relative, 598, 599, 601, 619 fluid, 10, 416 universal function, 606 Shear velocity, 350 wall, 595, 602 Similarity Roughness length, 453, 454 complete, 616, 617 incomplete, 615, 617 Simple shear flow, 356 S Slide bearing, 391 Saddle point, 405 theory, 391 Scale analysis, 555 Snout, 398 Scaling Solenoidal, 226 time, 570 mass flux, 78 Scaling rule, 580 part, 176 Schwarz-Christoffel, 316 formula, 322, 325 vector field, 216, 221 map, 321 velocity field, 78, 216, 220, 489 mapping, 317, 321 Sommerfeld number, 396 transformation, 315–318, 320–323, 327, Source 330, 331 conservation (law), 280 type mapping, 327 flow, 286 Semi-convex, 264, 266 Source-free area, 264 solenoidal velocity field, 216 body, 264, 266 velocity field, 209 part, 266, 267 Specific heat Separation, 553, 557 at constant pressure, 407 critical region, 557 Speed of sound, 11, 85 flow, 507 Stability of equilibrium position, 40 instant, 558 analysis, 40 point, 553 explaining, 41 returning flow, 506 Stagnation flow time, 558 plane, 505 Separation of variables streamline, 533 Subject Index 637

Stagnation point, 71, 103, 104, 239, 256, 257, Strain rate tensor, 8 259, 260, 262, 263, 283, 287, 288, 294, Streakline, 64, 66, 67, 70 297, 312–314, 487, 488, 507, 549, 566 of a cigarette smoke, 68 flow, 256, 257, 281–284, 480, 505 Stream function, 71, 247, 249, 250, 253–256, flow, rotating, 508 332, 468, 469, 500, 502, 503, 524, 533 flow, rotation-symmetric, 495 of a barotropic fluid, 250 flow, three-dimensional, 495 of a three-dimensional flow field, 247 flow, three-dimensional elliptical, 495 Streamline, 64, 65, 67–72, 97, 103, 130, 174, flow, two-dimensional, 501 248, 278, 279, 281–284, 286, 292, 308, inviscid potential flow, 488 309, 311–314, 331–334, 470, 471, 488, three-dimensional boundary layer flow, 494 489, 495, 533, 534, 609, 611 two-dimensional, 488 filament, 68, 95 Stagnation pressure, 103, 143, 488, 490, 495, stagnation, 103 593, 609 tube, 68 Stagnation streamline, 103 Stress deviator, 8 Stagnation tube, 104 Stress-heating, 409 Statics, 2 Stress power, 409 Stationary, 5, 64, 235, 554 Stress-strain rate law, 360 axisymmetric laminar jet, 467 Stress-strain rate relation, 356 flow, 378 Stress-stretching relation, 356 plane film flow, 388 Stress (tensor), 351, 353, 366, 390, 391 plane flow, 387 Cauchy, 149–152, 161, 162, 351, 398 plane jet, 79 , 161 process, 64 dissipative viscous, 168 Steady extra, 351, 353, 444 condition, 117 fictitious, 604 Steady process, 64 fictitious shear, 604 Steady state, 5, 371, 499, 513 first Piola-Kirchhoff, 124 boundary layer thickness, 512 frictional, 444 condition, 75, 78, 106, 119, 152, 406 normal, 20, 21, 148, 149, 151, 359, 390 exit velocity, 113 octahedral shear, 371 flow, 106, 132, 390 Reynolds, 580 heat equation, 50 second Piola-Kirchhoff, 124 layer flow, 426 shear, 4, 5, 8, 10, 21, 84, 129, 148, 149, momentum equation, 500 151, 350, 366, 368, 372, 373, 375, 379, NS-equation, 508 381, 383, 386, 393–395, 401, 444, 451, pipe flow condition, 101 461, 506, 580, 582, 589, 603 process, 98 surface, 17, 18, 453 solution, 415 tensile, 20 symptotic, 512 turbulent shear, 450, 603, 604 Stokes turbulent viscous, 580 approximation, 387 viscous, 169, 349, 353, 359, 360, 366, 371, assumption, 360, 398 374, 390, 398, 400, 418, 498, 505 equation, 419 viscous frictional, 353 hypothesis, 6 wall shear, 507, 541, 543, 544 integral theorem, 163, 227, 228 wind shear, 446, 447 law, 199, 204, 205, 218, 228, 267 yield, 8, 9 theorem, 138, 139, 163, 169, 171, 172, 203, Stretch tensor, 181 205, 267, 269 left, 179, 185 Strain rate, 179, 356, 371 right, 179, 185 in cylindrical coordinates, 382 Stretching tensor, 178, 353, 354, 359, 366 tensor, 178, 182, 353, 354, 366, 450 Stretching transformation, 530, 531, 534, 535 volume, 359 Subcritical flow, 131, 132, 135 638 Subject Index

Sublayer fully developed rough, 601 laminar, 614, 617 plane, 602 viscous, 615, 617, 618 Turbulent kinetic energy, 580 Supercritical flow, 131, 132, 135 specific, 455 Surface flow density, 410 Turbulent mixing length, 580, 604 Surface force, 19, 20, 85, 89, 90, 117, 148, 149, 151, 351 per unit surface area, 19 U specific, 20, 148 Unit normal vector, 149, 218, 219, 242, 264, Surface tension, 17 266, 267 Surface wall layer, 455 exterior, 20, 21, 26, 90, 118, 149, 397, 401 System oriented, 201 absolute inertial, 47 external, 21 accelerated, 47 Unsteady flow, 459 inertial, 47, 48, 87 by a velocity jump, 462 non-inertial, 47 by an oscillating wall, 460

V T Velocity Taylor absolute, 48 series, 54, 211, 257 fixed body, 48 series expansion, 24, 53, 54, 101, 300, 303, of a particle, 61 383, 416, 448 relative, 48 Taylor-Proudman theorem, 186 virtual, 48 Technically rough, 602 Velocity height, 610 Temperature, 3, 50, 360, 363, 367, 409 Virtual absolute, 3, 52, 101, 407 body, 260 atmospheric, 408 body force, 236 empirical, 3 mass, 246 Kelvin, 3, 407 mass concept, 238, 245 melting, 411, 414 mass for sphere, 241 Thermal equation of state, 11, 26, 50, 97, 98 mass, generalized concept, 244 for ideal gas, 52, 100 mass, tensor, 244 Thermodynamic equilibrium, 3 velocity, 312 Thermodynamics, 3 volume of a sphere, 238 first law, 3, 407 Viscometer second law, 3, 398, 413 cone plate, 383, 384 Thin film deformation, 429 Couette, 379 Torricelli formula, 106, 107, 587, 589 Engler, 592, 619 Total transport, 448 falling sphere, 358 Traction, 21, 461 Viscosity, 6, 358, 359, 364, 368, 370, 379, 396, basal shear, 412 439, 505, 528, 587, 615 normal, 21, 148 bulk, 6 shear, 4, 5, 21, 149, 350, 461, 513, 517 dynamic, 6, 356, 357 surface, 161, 401 dynamic bulk, 8 wall shear, 506 dynamic shear, 5, 8, 9, 350, 363 Trajectory, 64–68, 70–73 eddy, 450, 455–457, 580 Trough, 405 effective, 366–369, 383 point, 405 for pure water, 362 Turbulent correlation, 580 kinematic, 6, 7, 145, 455, 474, 529, 589, Turbulent dissipation, 580 592, 605, 614 specific, 455 kinematic shear, 365 Turbulent flow, 579 shear, 6, 10, 360, 362, 365, 379, 593, 604 Subject Index 639

turbulent, 450, 580, 604 Von Kármán-Cochran volume, 360, 363 profile, 512 Viscosity law steady state velocity profile, 513 dynamic, 379 Von Kármán-Trefftz finite, 369, 370, 418 conformal mapping, 298, 302 infinite, 369–371, 418 mapping, 301 Viscous flow profile, 301, 302 along a flat, 507 transformation, 302 along a slender body, 507 Vortex around a circular cylinder, 558 -free, 166 around a corner, 503 filament, 170, 185 in convergent channel, 477 surface, 170 in divergent channel, 479 tube, 170–172 induced by a rotating disk of finite extent, Vortex sheet, 145, 170, 554 508 Vortex-free induced by a rotating disk with infinite flow field, 226 extent, 507 irrotational field, 209 inside a wedge, 505 velocity field, 209 past a corner, 503 Vorticity, 25, 98, 146, 162, 166, 172, 209, 220, past a semi-infinite plate, 533 231, 499 past an infinite wedge, 500 absolute, 167, 186, 190, 193 Viscous fluid, 6, 146, 155, 181, 349, 350, 361, baroclinic potential, 193, 194 366, 378, 383–385, 397, 405, 414, 425, barotropic potential, 192 427, 438, 498, 542, 587, 588, 604 equation, 175, 177, 190, 191, 533 around a corner, 503 filament, 219 density preserving, 381, 387 per unit mass, 177, 178 due to a rough rotating disk, 512 planetary, 167, 168 in a vessel, 586 potential, 190, 193, 194 linear, 7, 371, 373, 585, 586 relative, 168, 193 Newtonian, 360 source, 220 non-linear, 8 tensor, 178, 182 past a semi-infinite length plate, 516 vector, 183, 250 past a semi-infinite plate, 487, 517 Vorticity theorem, 169 through a circular conduit, 430 Ertel, 188, 190 through pipe, 579 Helmholtz, 175, 177, 185, 186, 188, 190, Volume flux, 101, 116, 124, 133, 377, 385, 250 435, 527–529, 541, 591 potential, 188–190 between the streamlines, 249 specific, 528 Vorticity/strength, 217 vertically integrated, 447 Volume force, 19, 20, 86, 89, 90, 117, 148 specific, 83, 90, 119 W Volume preserving, 10 Wall bounded shear/layer flow, 349, 356, 357, fluid, 78 371, 372, 379, 383 Volume transport, 449 through a slot, 372 Von Kármán Wall shear velocity, 604 constant, 450, 605 Walz’s quadrature formula, 551, 552 formula, 601 Water line, 40, 42–46 steady profile, 512 Wave number, 461 vortex street, 69, 553 Wind factor, 455