R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 89
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Bernoulli, is extended to nonsteady flow and given a satisfactory derivation by J. Bernoulli.-The equation was generalized by Euler in the papers quoted inNote 3; Euler also gave explicitly the equation for a streamline. In the same papersappears the integral Jdp/p. J. Bernoulli originated the concept of hydraulic pressurewhich was later generalized by Euler (Note 3). Regarding the quoted works of theBernoullis see [13] p.
L X X X I V IT., and the chapter on "Bernoullian Theorems",TRUESDELL [11], p. 125 if.11. Flows throughout which Η is constant are often called "isoenergetic". (The466NOTES A N DADDENDAArticle 2term " h o m e n e r g i c " is used by L . HOWARTH [24].) Compare in this connection Sec.24.1, and N o t e V . 41.Article 312. T h e fundamental concept of stress which is at the basis of today's mechanics ofcontinua is due to A .
L . Cauchy (1789-1857); it generalizes Euler's hydrostatic pressure: A . L . CAUCHY, "Recherches sur l'oquilibre et le mouvement intorieur des corpssolides ou fluides, olastiques ou non elastiques", Bull. soc. philomath. Paris (1823),pp. 9-13, a brief summary; and in full detail: " D e la pression ou tension dans un corpss o l i d e " , Oeuvres Completes, Ser.
2, V o l . 7, Paris: Gauthiers-Villars, 1889, pp. 60-78;other papers by Cauchy on this subject are cited in TRUESDELL [12], p. 264, as 1827.3and 1828.1. T h e basic properties of the stress tensor Σ, its symmetry, transformationformulas, etc., are easy to find on the first pages of most textbooks on elasticity.T h e formal theory of Σ is of course the same no matter whether elasticity theory,the theory of viscous flow, or theory of plastic flow is considered, while the physicalassumptions are quite different in the various fields (cf.
the title of Cauehy's 1823paper, cited a b o v e ) . On these fundamentals cf. also R. v. M I S E S , " U b e r die bisherigenAnsatze in der klassischen Mechanik der K o n t i n u a " , Verhandl. S. intern. Congr. tech.Mech. Stockholm 2 (1930), pp. 3-13.13. T h e concept of fluid friction being proportional t o the relative gliding of neighboing layers goes back to N e w t o n , second book of the Principia,in the " H y p o t h e s e s "preceding Proposition L I . L . N A V I E R , ["Momoire sur les lois des mouvements desfluides", Mem. acad. sci.
Paris 6 (1822), p. 389ff.] developeda corpuscular theory thatled to the still accepted system of partial differential equations for incompressibleviscous fluids. A different corpuscular theory leading to the accepted equationswithout restriction to the incompressible case was given by Poisson. Saint Venant(1843) and particularly G. G.
Stokes (1£45) derived the same equations on the basisof the general stress concept for a continuously distributed mass: G. G. STOKES," O n the theories of the internal friction of fluids in motion, and of the equilibriumand motion of elastic solids", Trans. Cambridge Phil. Soc. 8 (1845), p. 287 ff.
(Compare also v. M I S E S [16], p. 615; LAMB [15], p. 652; BUSEMANN [19], p. 351.) In thepresent chapter the usual simple form of the dependence of the viscous forces uponthe variables p, p, q is not assumed, see p. 26 ff. and also end of Sec. 3. Actually viscousflow is not studied in our book. Compare however Arts. 11, 14, and 22.14. T h e tensor symbol, grad Σ (in the sense of W . G i b b s ) , can be understood byconsidering it as the product of the symbolic vector " g r a d i e n t " (see p. 3) multiplied by the tensor Σ according to the rules of matrix multiplication.
This exhibitsthe vector character of grad Σ.15. T h e invariant character of — w' may be seen in the following way. T h e threeexpressions in parentheses in E q . (10) are the components of the vector u = S'»q,this product being computed by matrix multiplication. T h e right side of E q . (10) isthen the divergence of this vector u, viz., —vo' — div ( Σ ' - q ) .16.
θ can vanish only if either D or Σ' vanishes (cf. N o t e 17).17. T o see the invariant character of Θ, we form by matrix multiplication thetensor a = Σ'·Ό\ then θ = a + a + o « . Such an expression, the " t r a c e " of anη Χ η matrix, is invariant; cf. E q . ( 4 ) .18. As in N o t e 15: -w' = div (S'»q) ss div u.
N e x t , (2.27) is applied to / d i v u dV,•and it is then easily verified that u = t' *q, as in (20).19. This general form of specifying equation was introduced and discussed byv . M I S E S , cit. N o t e 6. Our E q . (8') is replaced there by dq/dt + (1/p) grad ρ = F. InxxyvKnnCHAPTER467IArticle 3the " i d e a l " case of our text (see N o t e 8) A, B, C,F,are given functions of the ninevariables r, t, q, ρ, ρ not involving derivatives.
In a more general case certain firstand second-order derivatives of q, ρ, p are also admitted in A, B, C, F.Article 420. In this book magnitudes corresponding to a state of rest are in general denotedby the subscript " s " , meaning " s t a g n a t i o n " as in "stagnation p o i n t " , "stagnationpressure", etc. H e r e , however, the subscript " 0 " (zero) is used. This is becausethe subscript essentially serves to denote undisturbed flow, to which a perturbation is applied, and this undisturbed state is not necessarily a state of rest (see A r t .5 ) . In the case of acoustics, the unperturbed state denotes a gas (air) at rest; in aerodynamical applications, however, we rather consider the small perturbation of aparallel flow. A t the basis of small perturbation theory is, likewise, the wave equation.T h e notation in A r t .
4 is adapted to that in A r t . 5.Small perturbation theory, which plays an important role in aerodynamics, is notconsidered further in this book. W e mention the following references: G. N . W A R D ,LinearizedTheory of High-SpeedFlow, London and N e w Y o r k : Cambridge U n i v .Press, 1955; the condensed monograph, S. GOLDSTEIN, " L i n e a r i z e d T h e o r y of Supersonic F l o w " , prepared by S.
I . P A I , Inst, for Fluid Dynamics andAppl.Math.,Univ.of Maryland,Lecture Ser. No. 2 (1950), I . IMAI, "Approximation methods in compressible fluid dynamics", ibid., Tech. Note BN-96(1957), and the article by W . R.SEARS, "Small Perturbation T h e o r y " , in [31], pp. 61-121. See also Notes 111.46,V. 24, 64.21. In the famous paper, J . L. D'ALEMBERT, "Recherches sur la courbe que formeune corde tendue mise en v i b r a t i o n " , Kgl. Akad. Wiss. Berlin 3 (1747), p. 214 ff.
theauthor treats the string as a continuous medium. This theory was refounded and thoroughly exploited by Euler. In another treatment Euler and Lagrange imagined thestring made up of a finite number of equally spaced particles and performed thepassage to the limit, see J. L.
LAGRANGE, "Recherches sur la nature et la propagationdu s o n " , MiscellaneaTaurinensia1 (1759), pp. 1-112.22. T h e first attempt at a mathematical theory of sound was made by N e w t o n .B y a very devious argument he reached the formula αο = \/po/po , thus obtainingthe same result as if he had assumed the motion isothermal. I n the 1687 edition of thePrincipia,he obtained the value 968 ft/sec; in the 1713 edition he got 979 ft/sec. T h etreatment based on the wave equation with ρ = Kp is due to d'Alembert and Euler.T h e possibility of a computation based on a different (p, p)-relation was suggestedoccasionally in the 18th century. Physical arguments in favor of ρ = Kp —withthecorresponding value a = VrPo/po—were first advanced by J .
Β . B I O T , (1774-1867)["Sur la theOrie du s o n " , J. phys. 55 (1802), pp. 173-182] who acknowledged the assistance of LAPLACE (1749-1827) [the result was later included in P. S. LAPLACE, Traite demecanique celeste, Vol. 5, Paris: Duprat (1825)]. T h e modern explanation based onthe concept of an adiabatic process did not become possible, of course, until afterthe creation of the mechanical theory of heat in the 19th century.T h e mathematicians d'Alembert, D . Bernoulli, Euler, and Lagrange discovereda large part of the theory of production and propagation of sound.
T h e theory washighly developed in the 19th century by S. D . Poisson (1781-1840), G. G. Stokes (18191903) and particularly by H . v. Helmholtz (1821-1894), G. Kirchhoff (1824-1887),and Lord Rayleigh (1842-1919). H . v. HELMHOLTZ, " D i e Lehre von den Tonempfindungen als physiologische Grundlage fur die Theorie der M u s i k " , published 1862,Braunschweig: F. Vieweg, 1913. (English translation: On the Sensations of Tone as ay0468NOTES A N DADDENDAArticle 4Physiological Basis for the Theory of Music, reprinted 1954, N e w Y o r k : D o v e r ) . LORDRAYLEIGH'S Theory of Sound [28], is a work which still offers unexhausted treasures.(Compare also the instructive historical introduction to Rayleigh's work by R . B.Lindsay).23.
S . D . POISSON, "Sur lemouvement des fluides olastiques dans les tuyaux cylindriques, et sur les theories des instruments a v e n t " , Mem. acad. sci. Paris, S6r. 2, 2(1817), pp. 305-402. Poisson's method, as given in our text, is the one adopted byLORD RAYLEIGH [28], Vol. I I , Sec. 273 ff.24. T h e method used in this section is Hadamard's "mothode de doscente", see J.HADAMARD, Lectures on Cauchy's Problem in Linear Partial DifferentialEquations,Paris: Hermann, 1932, p. 49, (reprinted, N e w Y o r k : Dover, 1952).A very interesting paper on the wave equation with discussion of the basic difference between odd and even η (in our case η = 1 and η = 3 versus η = 2) is M .
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