R. von Mises - Mathematical theory of compressible fluid flow, страница 103

Press, 1954. Also a forthcoming volume in the sameseries: EMMONS, H . W . (editor), Fundamentalsof Gas Dynamics, 1957.32. Shapiro, A . H . , The Dynamicsand Thermodynamicsof CompressibleFluidFlow,2 vols., N e w Y o r k : Ronald Press, 1953.33. Weber,H . , DiePartiellenDifferentialgleichungender mathematischenPhysik(Nach Riemann's Vorlesungen bearbeitet von H . W e b e r ) , 5th ed., Braun­schweig: F. Vieweg, V o l . I I , 1912.TABLES34.

Aeronaut. Research Council; Tables for Use in CalculationsAirflow, London and N e w Y o r k , Oxford U n i v . Press, 1954.35. D a i l e y , C. L., and W o o d , F. C , ComputationofCurves for CompressibleCompressibleFluidProb­lems, N e w Y o r k : W i l e y , 1949.36.

Emmons, H . W . , Gas Dynamics Tables for Air, N e w Y o r k ; D o v e r , 1947.37. Handbook of Supersonic Aerodynamics,N A V O R D Report 1488, 6 vols., Washing­ton, D , C . : U . S. G o v t . Printing Office, 1950-,AUTHORINDEXNumbers in italics refer t o the text pages 1-463, non-italic numbers t o the Notesand Addenda, i.e. pp.

464-503.Clauser, M . U., 486Cole, J . D . , 499Copson, Ε . T . , 477, 479Courant, R., 473, 474, 479, 481, 486, 487,502, 503Craggs, J. W . , 487, 490, 491Crocco, L., 426, 482, 496AAckeret, J., 494, 498, 503Amontons, G., 465Aschkenas, H . , 499ΒBarnes, E. W . , 491Bateman, H . , 470, 483, 503Bechert, K . , 478, 497Becker, R., 147, 475, 476Beckert, H . , 474Beltrami, E., 469DDailey, C. L., 503d'Alembert, J . L., 85, 86, 465, 467, 469Darboux, G., 474, 476, 477, 484Davies, H .

J., 489de L a v a l , C. G. P., 445Demtchenko, B., 483de Prima, C , 479de St. Venant, B., 466, 471Diaz, J. B., 493Dibble, C. G., 490Dryden, H . L., 503Duhem, P., 472, 481Durand, W . F., 503Bergman, S., 851, 362ff, 872ff, 442, U4,474, 488, 491, 492, 493, 497, 502Bernoulli, D . , 465, 467Bernoulli, J., 464, 465Bers, L., #0, 485, 498, 500Bethe, Η .

Α . , 481Betz, Α . , 480Bickley, W . G., 484Biot, J. B., 467Birkhoff, G., 490, 494Bleakney, W . , 495Bousquet, Μ . M . , 465Boyle, R., 465ΕBusemann, Α . , 95, 260, 459, 461, 466, 471,483, 494, 495, 500, 503CCarrier, G. F., 503Cauchy, A . L., 120, 126, 247ff, 269, 298,SOSff, 466, 469, 470, 486Chang, C.

C , 490Earnshaw, S., 479Emmons, H . , 452, 499, 503Epstein, B., 492Epstein, P . S., 495Euler, L., 4, 6, 464, 465, 466, 467, 469,476, 477FFerguson, D . F., 490, 497Ferri, Α . , 471, 490, 495490, 493, 498Finn, R., 488, 498Charles, J. A . C , 465Flanders, D . Α . , 499Cherry, Τ . M . , 827, 861, 852, 448ff, 446ff, Frank, P., 501462, 488, 490, 493, 497Frankl, F.

I . , 469, 460, 461, 490, 500Clauser, F. H . , 485, 486Friedmann, P., 470Chaplygin, S. Α . , 251, 266ff, 278ff, 829ff,846ff, 851ff, 44$, 483, 484, 485, 488, 489,504AUTHORFriedrichs, K . O., tfSff, 462, 473, 479,480, 481, 482, 486, 492, 494, 497, 499,503GGalileo, 464Garrick, I. E., 489Gauss, K . F., 21, 488, 489Geiringer, H., 473, 482, 487Gelbart, Α., 485Gellerstedt, S., 460, 500Germain, P., 488Gibbs, W., 466Gibson, W., 486, 493Giese, J. H., 486Gilbarg, D., 475, 476, 480, 488, 498Gilles, Α., 494Glass, I . I., 482Glauert, H., 452, 498Gortler, H., 497, 498Goldstein, S., 467, 491, 497Goursat, E., 489Guderley, G., 459, 461, 483, 500Guenther, P.

E., 496ΗHaack, W., 474Hadamard, J., 457, 468, 469, 471, 472,474, 496, 499, 502Hamel, G., 483, 488, 496Hankel, H., 469Hansen, A . G., 496Hellwig, G., 474Hencky, H . , 483Henning, H., 468Hicks, B. L., 496Hilbert, D . , 474, 502Hopf, L., 494Howarth, L., 465, 466, 503Huckel, V., 489, 497Hugoniot, H., 201, 207, 479, 480, //81, 487Huygens, C., 464IImai, I., 467JJaeckel, K., 485505INDEXJeans, J.

H., 476Joukowski, N . , 847, 498ΚKahane, Α . , 495Kampo de Foriet, J., 488, 502Kaplan, C., 489, 498, 499K e l v i n , L o r d : see W. ThomsonKibel, I . Α . , 470Kirchhoff, G., 847, 849, 465, 467K l e i n , F., 488Kolodner, I. I., 455, 499K o p a l , Z., 470Kotschin, N . J., 470K r a f t , H., 490K u e r t i , G., 497K u o , Υ . H., 458, 487, 491, 493, 497, 498,499K u t t a , W.

M . , 498LLaby, Τ. H., 476Lagrange, J. L., 4, 464, 467, 469, 479L a m b , H., 465, 466, 470, 502Lampariello, G., 472Lanchester, F. W., 481Laplace, P. S., 89, 459, 467Lax, P., 474, 480Lees, L., 495Legendre, A . M . , 261ff, 484L e Roux, J., 493L e v e y , H. C., 496, 497Levi-Civita, T . , 471, 472, 473, 501L e w y , H., 473, 474L i b b y , P. Α., 475Liepmann, H. W., 499Lighthill, M . J., 827, 351ff, 871, 872ff, 442,444, 446ff, 452, 475, 480, 481, 485,487,488, 489, 491, 493, 494, 495, 497, 499Lin,C.

C., 285ff, 485Lindelof, E. L., 489Lindsay, R. B., 468, 503Lipschitz, R., 476L o v e , A . E., 477, 478, 479Ludford, G. S. S., 475, 478, 482, 487, 492,493, 496Lussac, G., 465ΜMaccoll, J . W., 470, 475, 48650GAUTHORMach, E., 48, 464, 468, 495, 502Mach, L., 468Mackie, A. G., 477Mandel, J., 487Man well, A . R., 455, 499, 500Mariotte, E., 465Martin, Μ . H., 482, 496Massau, J., 259, 473Meyer, R. E., 486, 487, 488M e y e r , T .

, 468, 485, 494, 497Millikan, R. Α., 475Milne-Thompson, L. M . , 485Mittag-Leffler, M . G., 860Molenbroek, P., 483Monge, G., 471, 472Morawetz, C. S., 455, 500Morduchow, M . , 475Munk, M . , 483, 496Murnagham, F. P., 503ΝNavier, L., 186, 197, 225, 466, 476Nelson, Ε.

Α., 476Newton, I., Iff, Sff, 464, 465, 466, 472Nikolskii, Α. Α., 455, 457, 500ΟO'Brien, V., 489, 490Oswatitsch, K., 483, 503ΡPack, D . C., 479, 490Pai, S. I., 467Paolucci, D., 476Patterson, G. N., 482Pe>es, J., 483Pfriem, H., 481Pidduck, F. B., 477, 478, 479Pillow, A . F., 479, 481Platrier, M . C., 479Poisson, S. D., 40ff, 466, 467, 468, 476,479Polachek, H., 482, 495Prandtl, L., 95, 148, 452, 468, 471, 475,480, 481, 483, 485, 494, 498Prim, R. C., 496Protter, Μ . H., 500RRankine, W . J. M .

, 201, 475, 480INDEXRayleigh, Lord (J. W . Strutt), 149, 467,468, 475, 479, 480, 503Riabouchinsky, D . , 483Richter, H., 495Richtmeyer, R. D . , 482Riemann, G. B., 126ff, 201,211ff, 470, 471,474, 476, 477, 478, 479, 480, 481, 487Riesz, M . , 468, 474Ringleb, F., 885ff, 451 ff, 454, 484, 486,487, 488, 489Rose, N . W., 470Rudenberg, R., 480Runge, C., 473sSalcher, L., 468Sauer, R., 473, 474, 476, 478, 483, 484,487, 488, 501, 503Schiffer, M . , 473, 474, 491, 492, 493, 502Schot, S. H., 487, 488Schubert, F., 494Schultz-Grunow, F., 478Sears, W .

R., 467, 468, 498, 499, 503Seeger, R. J., 482, 495Shapiro, A . H., 503Shiftman, M . , 450, 498Sommerfeld, Α., 470, 471, 502Steichen, Α., 483Steinberg, L., 486Stocker, P. M . , 488Stokes, G. G., 186, 197, 225, 466, 467,469, 470, 476, 479Sutherland, W., 476ΤTaganov, G. L, 455, 457, 500T a i t , P. G., 469Tamada, K., 452, 497, 498Taub, A . H., 478, 479, 495Taylor, G. L, 189, 452, 470, 475, 486,497, 498Temple, G., 486, 489Thomas, L. H., 476Thomas, Τ .

Y . , 495Thomson, W . (Lord K e l v i n ) , 55ff, 61 ff,469, 470Toepler, Α., 468Tollmien, W., 483, 484, 486Tomotika, S., 452, 497, 498AUTHOR507INDEXTricomi, F. G., 449, 469ff, 471, 473, 485,498, 500Truesdell, C. Α., 464, 465, 466, 470, 496,502Tsien, H . S., 267, 278ff, 468, 454, 480, 485,487, 491, 497, 498, 499Ward, G. N . , 467Wasserman, R. H .

, 496Watson, G. N . , 488, 491, 502Weber, H., 474, 481, 503Wecken, F., 495, 496Weinstein, Α., 476W e y l , H., 481VWhittaker, Ε. T . , 488, 491, 502von Helmholtz, H., 65, 68ff, 220, 847, Willers, F. Α . , 473426, 467, 469, 470, 481, 483Williams, J., 489von Karman, T . , 267, 278ff, 458ff, 480,Wood, F. C., 503485, 486, 494, 498, 499, 502Wosyka, J., 495von Mises, R., 286, 291, 465, 466, 469,470, 471, 472, 473, 475, 477, 478, 480,Y481, 482, 485, 491, 492, 496, 498, 499,502, 503Yarwood, J., 486, 489von Neumann, J., 481, 482WWalsh, J . W., 494Wantzel, L., 471ΖZaldastani, O., 476Zarantonello, Ε. H . , 490SUBJECT INDEXNumbers in italics refer to the text pages 1-463, non-italic numbers to the Notesand Addenda, i.e. pp. 464-501.for flow in a plane duct, 249, 270, 271,A305ff,445ffAdiabatic (see also Isentropic, Specify­for flow past a profile, 851ff, 442ff, 449,ing equation)491, 492simply, 10, 877, 436ffmixed, 271, 808, 449, 459ff, 474, 500strictly, 10, 208, 209, 428, 424, 426,in one-dimensional flow, 172481, 488, 441, 465, 482for subsonic jet, 846Adjoint equation, 127, 167Branch line, 812ff, S25ff, 446ff, 487Affine transformation, 131, 185double branch point, 827, 448Asymptotic solution, 219, 380, 441ff, 480,496cAxially symmetric flow, 81, 88, 85, 465,482, 483, 496Cauchy's equations, 470Cauchy problem, 120, 126, 247ff, 269,Β804ff, 486Bend, flow aroundconvex, 800concave, 804Bergman's integrationCauchy-Riemannmethod, 362ff,872ff, 491, 493Bernoulli constant (Bernoulli function;see also Head, t o t a l ) , 18, 65, 66,correspondence, 288ff, 882ffequation (for stream function), 251266, 880, 854, 449, 485, 488ffflow, 288fffunction, 381Bernoulli equation, 17ff, 76, 141, 205,240, 281, 829, 424, 427ff, 435in differential form, 18, 889Boundary conditions, 6, 196, 211, 226,849,411, 486, 449279,Channel flow, 445ff, 470supersonic, 805fftransonic, 448ffChaplygin87, 90, 875, 888, 428ff, 430270, 271,equations, 251,282, 288, 884352, 875,402, 406,j e t , 346ffmethod, 829ff, 348, 851, 362, 493Chaplygin-Kdrman-Tsien approximation(see also Linearized condition),initial conditions, 6, 226267, 278ffBoundary layer, 442, 480, 496, 497CharacteristicBoundary-value problem (see also Ini­cone (see Mach cone)tial-value problem)coordinates (variables), 166, 265, 477,Cauchy problem, 120, 126, 247ff, 269,304ff483direction, 159, 161, 245, 346, 473for channel flow, 851ff, 445ffexceptional direction, 106, 817characteristic, 122, 172, 176, 247, 259,exceptional plane, 818, 820, 472269, 808, 474line, 108, 281correctly set, 457, 460quadrangle, 190, 808existence and uniqueness theorems for,triangle, 211, 875, 473120ff, 802ff, 449ff, 457, 461508SUBJECTCharacteristics (see also Characteristicline, Mach line, Compatibility re­lation)cross- 181, 182, 288, 294ff, 307ff, 328405, 408, 458forgeneral nonsteady nonpotentialflow, 112in the hodograph plane, 252ff, 287ff446ff, 484-for a linear system, 117ff, 162for one-dimensional nonsteady509INDEXDD ' A l e m b e r t ' s solution, 86ffDiscontinuous solution, 200, 219, 380,471, 472Dissipation function, 28D i v (see also Gauss' theorem = Diver­gence theorem), 21, 466Doublet (line doublet), 344S, 490Dupin's indicatrix, 92ffflow,112Εof pairs of equations, 116ff, 473plus and minus, 245Edge (of regression), 134, SI3, 826ff, 447,of a second-order equation, 107, 158,487429, 471Elastic fluid (overall (p, p)-relation), 7,in the speedgraph plane, 162424for steady two-dimensional potentialenergy equation for, 16flow, 109expansion energy for, 17for steady three-dimensional potentialirrotational flow of an, 92flow, 110polytropic gas, 8of a system of equations, lOSffEllipticwith viscosity and heat conduction,equation (problem), 118, 132, 271, 362,472449, 460ff, 485Circuit, 55ff, 469point, 92Circulation (see also K e l v i n ' s theorem),Enthalpy, 1955ff, 285, 444, 469, 485, 498EnergyCompatability relationinternal, 14for nonsteady one-dimensional flow,for a nonperfect gas, 14ff112, 155ffkinetic, 13for pairs of equations, 119ff, 473potential, 13for plane steady potential flow, 246Energy equationfor second-order equation, 107ff, 265for an elastic fluid, 16for system of equations, 106, 472for an element of a nonperfect gas, 16Conical flow, 486for finite mass of a perfect gas, 22, 23Contact discontinuity, 221ff,406ff, 420,for a fluid element of a perfect gas, 14473, 481for an inviscid fluid, 11Contact transformation (see Legendrefor a perfect viscous fluid, 29, 136fftransformation)EntropyCorner, flow around, 298ff, 336, 341ff,change across a shock, 202ff, 382, 481,402, 486496Critical curve, 818, 816ff, 886, 846distribution in nonisentropic simpleCurl, 57, 4243, 469wave, 234and mean rotation, 60in nonisentropic flow, 423ffStokes' theorem, 58of a nonperfect gas, 14, 15and vortex vector, 60of a perfect gas, 8CuspEpicycloid, 254ff, 298at limit line, 817ff, 486Equation of continuity (see also N e w ­of limit line, 819ff, 486ff, 489ton's Principle), 5ff, 135, 199, 288,of stream line, 486Cylindrical w a v e , 86, 112374, 376, 430, 433510SUBJECTEquation of motion (Newton equation,Euler's equations; see also N e w ­ton's Principle)for an inviscid fluid, 4, 85, 89, 280,874, 876, 424, 4S0, 488for a viscous heat-conductingINDEXplane, 91, 252, 256, 267, 824, 856, 884,419, 446potential, 884, 870singularity, 81 Iffsolution, 268ff, 812, SSlff, 885, 444, 462fluid,26, 88, 185ff, 199 ·Equation of statefor perfect gas, 8, 465for nonperfect gas, 14Euler's equations, 6, 465Euler-Poisson-Darboux equation, 476Euler's rule (see Material differentia­tion)Exceptional (see Characteristic)Exceptional direction, 817ff, 824FFirst L a w of Thermodynamics, 9, 424Flow past,circular profile (circular cylinder),856ff, 442ffprofile, 442ff, 449ffsemi-infinite cone, 476straight line profile, 406ffstraightened profile, 456ff, 499ffFlow intensity, 89Folium of Descartes, 886ffFrankl's problem, 460ffGGauss' Theorem (Divergence T h e o r e m ) ,M,4iGradient, 8, 65, 69, 466ΗHeadpressure, 18, 19, 205, 888total, 18, 205, 888, 4US, 481, 485Heatconductivity coefficient, 21, 149, 875Helmholtz' equation (see also Helm­holtz' theorems), 67Helmholtz' theorems, 64ff, 470Hodographspace, 90transformation (method), 90, 95, 161,249, 470, 483, 486, 488ff, 497Hugoniot equation (curve), 207, 208, 884,481Hyperbolicequation (problem), 118, 120, 124, 182,159, 244, 271, 449, 461ff,476, 480,482, 493point, 92Hypergeometric function, 881, 841, 848,849 , 857, 477 , 488ff, 497Hypersonic, 49IIdeal fluid, 88, 165, 465Initial-value problem (see also Bound­ary-value problem, Boundary con­ditions), 168, 176, 474, 480for small perturbations, 42Intensity of propagation, 60ffInternal force, 2for an inviscid fluid, 8for a viscous fluid, 26Inviscid fluid, 8Irrotational flow, 59, 87, 157, 470and Bernoulli function, 66, 875of an elastic fluid, 92,plane steady, 95, 287ff, 827and potential, 70Isentropic (see also A d i a b a t i c ) , 8, 465non-, 229ff, 428ffIsobar, 90Isothermal, 8, 19, 97, 425JJacobian180characteristics, 262ff, 287ff, 400ff, 408ffΚ446ffequation (see also Chaplygin equation),250ff, 261ff,868, 493, 497mapping, 249, 287(Jacobian determinant), 180,K e l v i n ' s theorem (see also circulation),68ffSUBJECTLINDEX511Material differentiation (particle dif­ferentiation, Euler's rule of dif­ferentiation), 2, 8, 464Material line (filament)vortex line as, 64, 426M i x e d flow (see also Transonic), 88, 84,Lagrangian equations, 4de L a v a l nozzle (convergent-divergentnozzle), 445, 486, 497, 498Laplace equation, 71, 280, SS2, 459, 485operator, 89, 71276, 448ff, 460ff, 4595, 487, 492Legendre transformation, 262ff,A83fiMixed problem (see Boundary-valueLighthill's method, 852ff, 868, 870, 871,problem)872ff, 444Monge-Ampere equation, 282, 285, 485Limiting (limit)Monge cone, 471, 472circle, 88, 277Monotonicity law, 457cusp of limit line, 319, 444, 456, 486487, 489Νdouble limit point, 819, 889, 846, 445Navier-Stokes theory, 186, 225intersection of limit lines, 820N e w t o n equation (see Equation of mo­line, 184, 270, 81 Iff, 886ff, 448, 451,tion, N e w t o n ' s Principle)458ff, 486, 487, 499Newton's Principle (see also Equationsonic, 822ff 487, 488of m o t i o n ) , Iff, 464point, 817Newton's Second L a w (see alsn EquationLinear differential equation (s), 117ff,of m o t i o n ) , 1, 226124ff, 162ff, 250ff, 262ff, 829, 471,N o d a l point, 146474, 492N o d a l point ( = lattice point) of network,Linearized flow (small perturbation),Hff,46ff, 86, 452, 467, 494, 498Linearized condition (see also Chaplygin-Kdrm&n-Tsien g a s ) , 17, 19LocalMach angle 49Mach number 49sound velocity 49ΜMachangle, 48, 49, 882, 404, 468cone, 48, 114, 472line (characteristics), 52ff, 95, 102,108,108, 245ff,256,291ff, 805,810, 818, 818ff, 480, 468, 483net, 258ffnumber, 48, 49, 89, 142, 202, 208, 875,880, 881, 898, 418, 481, 448Mapping (see Hodograph)Massau's method, 259ff, 473Material discontinuity, 472Maximumcircle, 91, 254, 299, 887v e l o c i t y (speed), 89ff, 240ff, 888ff,Mean rotation, 60and Bernoulli function, 65Meunier's theorem, 93168, 247, 259, 260Nonlinear, 78, 108, 241, 271, 449, 461,474, 480, 500Nonperfect gas, 14, 481, 494energy equation for, 16entropy for, 14, 15equation of state for, 14internal energy for, 14, 15specifying equation for, 16Nonsteady flow, 86, 114one-dimensional, 78, 112, 155ffΟOne-dimensional flownonsteady, 78, 112, 185, 155ff, 829, 476,477, 481, 497general integral for, 165ffRiemann function for, 167, 474, 477,480steady, 187ff, 148ΡParticle function, 158, 160, 168, 186, 225,280Particle line (world l i n e ) , 5, 78, 79, 182,188, 211, 473as characteristic, 114, 220512SUBJECTPeres-Munk (p,p)-relation, 483Perfect gasdiatomic, 97, 476energy equation for, 22entropy of, 7equation of state for, 7, 465monatomic, 476pressure head, 19specifying equation for viscous, 30Physicalplane, 91space, 90Planar differential equation (see alsoLinear and Nonlinear), 103, 117,124, 130, 157, 281, 248, 429, 436, 471Plane flow (motion), 9, 465radial, 77steady, 424, 426, 489, 481steady irrotational, 237ffsteady plane flow of viscous fluid,377ff, 440, 496Poisson's solution (Poisson's formula),40, 42ffPolytropic gas (see also Elasticfluid),8, 95, 156, 164ff, 286, 251,253ff, 291,INDEXPrandtl-Meyer flow (see also Simplew a v e ) , 485Prandtl number, 148, 149, 150, 441Prandtl's relation, 481, 494Pressuredynamic, 96hydraulic, 8, 24ff, 465hill, 92ff, 98, 257, 887ff, 494in an inviscid fluid, 3in a viscous fluid, 25RRadial flow (source and sink flow)plane (see also Cylindrical w a v e ) , 77,272, 883, 470, 486three-dimensional (see also Sphericalw a v e ) , 75Rankine-Hugoniot conditions, 480Reciprocal lattices, 261Riemann method (solution), 127ff, 474,478invariants, 476, 482Riemann problem, 211, 224, 228Ringleb flow, 885ff, 451ffRotational flow, 425, 429, 433, 454, 470829ffsκ = 1.4, 75, 97, 148, 161, 286, 492κ = — 1 (see Linearized condition)Saddle point, lJfi, 816, 822κ = 5/3, 157Second L a w of Thermodynamics, 80isothermal, 97Shape correction, 284ffPotential (potential function), 70, 71, Shock (shock line, shock front, shock100, 157, 168, 186, 248, 244, 449ff,surface), 197ff, 200, 219, 229, 876,455acceleration, 469complex, 251, 283, 834, 343, 856equipotential line (potential880ff, 899ff, 425, 473, 475, 481, 482,494, 497, 499conditions, 198ff, 876ff, 380ff, 480, 494line),242, 272, 274, 317, 823, 326, 327equipotential surface, 70, 77Potential equation (see also Potentialflow, P o t e n t i a l ) , 72fffor axially symmetric flow, 81, 88, 85for one-dimensional flow, 157for steady plane flow, 79, 100, 108, 241 ff246, 485, 490for three-dimensional steady flow, 110,472Potential flow (see also Potential, P o ­tential equation)breakdown of, 454ff, 499fftransonic, 442ff, 498ffconical, 486curved, 224ff, 423, 430decay of, 481deflection, 381, 494diagram, 21 Off, 387ffentropy change across, 202ff, 382, 481,496head-on reflection of 214S, 418ff,interaction (collision), 221ff,419ff, 481,495ffMach reflection of, 495oblique, 877ffnonexistence of rarefaction, 480oblique reflection of, 41 Iffpolar, 386, 890ff, 400, 494SUBJECTregular reflection of, 420, 495stability of, 495strength of, 207, 395strong, 896ff, 413sufficiency of shock conditions, 436fftransition, 135ff, 210, 876ff, 488ff, 476of an elastic fluid, 6general form of, 32, 466of an incompressible fluid, 6, 17linearized condition, 17, 278for nonperfect inviscid gas, 16for simply adiabatic flow, 38, 186, 199,velocity of, 200377, 378weak, 896ff, 399ff, 417for strictly adiabatic flow, 10, 30, 38,zero, 201, 381155, 874Simple w a v e , 180ff, 191ff, 287ff, 298, 328,899, 458, 479, 485, 486backwardand forward,182ff, 288ff,400ffcentered,513INDEX184ff, 213, 289ff, 298, 479,486,interaction with shock, 21 Iffnonisentropic, 233ffparticle line and cross-characteristic,Speedgraphplane, 162, 209, 212, 221transformation, 161, 168Spherical wave, 86Spiral flow, 81, 275, 322, 486Stagnationdensity, 76, 88, 90, 388point, 91pressure, 76, 87, 88, 90, 388183ffsound velocity, 78penetration (interaction), 193, 808ff,temperature, 388478, 480, 481Steady flow, 5rarefaction (expansion) and compresStokes' Theorem (see also Circulation), 58sion, 182, 299ffStrainstreamlineand cross-characteristic,energy,17294fftensor, 29Singularity (see also Branch line, L i m i tline), 488Sonic,48ff, 89ff, 118,Stream function (see also Particle funct i o n ) , 242ff, 268, 329, 333ff, 354ff,182426ff, 484,491,497circle, 91, 254equation (see Chaplygin equation)double branch point, 827modified, 430fflimit line, 83, 322ff, 487Streamlinelimit point, 322ff, 324line, 83, 322ff, 839, 346, 448, 457, 461,822,499point 49, 52, 822ff, 839 , 448, 488point of a straight stream line, 328, 339,448transonic, 49, 445ff, 450ff, 453ff, 488fT,497ff, 500constantentropy,behind a shock, 41 Iffas characteristics, 114, 430flow)Specifying equation (condition), 6in a simple wave, 405, 409244,359,249, 281, 286, 287, 313, 315,860ff,866ff,881, 883, 397,422, 450, 485, 487, 492, 497for adiabatic flow, 9fffor the case of viscositycontact discontinuity, 408, 420Subsonic, 48, 49, 89, 118, 142, 182, 203,stagnation value of, 73Source and sink (see Radialasdeflection on crossing a shock, 881, 390Stress, 3, 24, 27, 137, 46649, 112, 230conduction, 33826, 827, 352, 875, 424, 425,465, 473, 486entropy distribution, 436Sound velocity, 88, 76particle-wisel i n e ) , 4,computation of, 268, 399ff, 423ffsphere, 91for(see also Particle52, 242, 246, 249, 291, 294, 317, 320,and heat-Subsonic j e t (see Chaplygin j e t )Substitution principle, 431514SUBJECTSupersonic, 48, 49, 54, 89, 118, 142, 182,208, 244, Π9,277, 818, 327, 859,874, 381, 888, 897, 406ff, 422, 429,451, 483, 485, 487, 490, 495, 497ffΤTelegraphist's equation, 478Trajectory ( p a t h ) , 4Transitionconditions (see also Shock conditions),198, 876 , 879, 476flow for viscous fluid, 142ff, 146ff, 150,158, 154Transition ( = transonic), 90, 240, 449line, 461Transonic (see Sonic)Tricomiequation (problem), 449, 459ff, 485, 500gas, 485VVelocity (see also Subsonic, Sonic, Super­sonic)complex, 251, 884head, 18, 205, 882local sound, 49maximum, 89ff, 240, 888of sound, 88, 71stagnation sound, 78INDEXViscosity, coefficient of, 136, 149, 197,875, 438Viscous fluidincompressible, 466one-dimensional flow for a heat con­ducting, 135ff, 475stress in, 24ffVortexfilament, 61, 469flow, 271, 838, 451, 470, 486line, 60, 64, 425, 469sheet, 473theorems (see also Helmholtz' theo­rems, Helmholtz' equation), 68,66, 67, 470tube, 61vector, 60, 426Vorticity, 61, 426, 468wW a v e equationgeneralized, 86in one dimension (D'Alembert's solu­tion), 86, 467in three dimensions (Poisson's solu­tion), 40, 467in two dimensions, 45Work, ISagainst viscous forces, 28.