R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 93
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A84 (1910), pp. 247-284, orScientific Papers, V o l . 5, London and N e w Y o r k : Cambridge University Press, 1912,pp. 573-610. As he (and later Becker, cit. N o t e 6) pointed out, the case μο = 0 is somewhat irregular; it must be treated as a limit, see D .
GILBARG, " T h e existence andlimit behavior of the one-dimensional shock l a y e r " , Am. J. Math. 73, (1951), pp.256-274. See also M . J. LIGHTHILL, " V i s c o s i t y effects in sound waves of finite amplit u d e " , Surveys in Mechanics, London and N e w Y o r k : Cambridge University Press,1956, pp. 250-351.04.
T h e complete problem was first fully treated by R. v. M I S E S , " O n the thicknessof a steady shock w a v e " , J. Aeronaut. Sci. 17 (1950), pp. 551-555. For a discussion ofthe nonperfect gas see D . GILBARG, N o t e 3.5. L. PRANDTL ["Eine Beziehung zwischen Warmeaustausch und Stromungswiderstand der Flussigkeiten", Physik. Z. 11 (1910), pp. 1072-1078] used this ratio in ahydrodynamical analogue of a heat transfer problem.6.
R . BECKER, "Stosswelle und D e t o n a t i o n " , Z. Physik. 8 (1922), pp. 321-362[translation: Ν AC A Tech. Mem. 506 (1929)]. Becker also considered the cases μ = 0and k = 0, see N o t e 3.7. LORD R A Y L E I G H , "On the viscosity of argon as affected by temperature", Proc.Roy. Soc. 66 (1900), pp. 68-74, or Scientific Papers, Vol. 4, London and N e w Y o r k :Cambridge U n i v . Press, 1903, pp. 452-458.8.
R . A . M I L L I K A N , " U b e r den wahrscheinlichsten Wert des Reibungskoeffizientender L u f t " , Ann. Physik 41 (1913), pp. 759-766. Millikan obtains his result by fittingSutherland's formula in the kinetic theory of gases to experimental values for air:0476NOTESA N D ADDENDAArticle 11W . SUTHERLAND, " T h e viscosity of gases and molecular f o r c e s " , Phil.
Mag., Ser. 5,36 (1893), pp. 507-531. I t is now generally accepted that the constant 223.2 in E q .(43) should be replaced by one closer to 200 (see for example [37], Vol. 5, p. 1504.1-1),but this has no effect on the conclusions.9. Τ . H . L A B Y and E. A . NELSON, " T h e r m a l conductivity; gases and v a p o r s " ,International Critical Tables, V o l . 5, N e w Y o r k : M c G r a w - H i l l , 1929, pp. 213-217. T h eformula is accurate in the range —312°F to 415°F.10. This is Eucken's formula, see J . H .
JEANS, Kineticand N e w Y o r k : Cambridge Univ. Press, 1952, p. 190.Theory of Gases,London11. For all but quite weak shocks this thickness is of the same order of magnitudeas the mean free path. Becker, cit. N o t e 6, questioned whether in these circumstancesthe equations of continuum mechanics are applicable to the problem, and the beliefhas grown that only kinetic theory is capable of a correct account of the transition.However several authors, starting with L. H . THOMAS [ " N o t e on Becker's theory ofthe shock f r o n t " , J. Chem. Phys. 12 (1944), pp.
449-453], have emphasized the considerable increase in thickness resulting from more realistic assumptions such astemperature dependence of viscosity and thermal conductivity. For a critical discussion and bibliography of the controversy see D . GILBARG and D . PAOLUCCI, " T h estructure of shock waves in the continuum theory of fluids", / . Rational Mech. Anal.2 (1953), pp.
617-642. These authors also investigate the effect of other viscosity assumptions than that of Navier-Stokes, E q . ( 6 ) .Article 1212. As was pointed out in N o t e 1.8, a distinction is made between the terms " i d e a l "and " p e r f e c t " . In this article we are primarily concerned with an ideal perfect gas inisentropic motion.
However the discussion is carried through for a general idealelastic fluid, with the polytropic case for illustration.13. For a general survey and useful bibliography of one-dimensional nonsteadyflow, see O. ZALDASTANI, " T h e one-dimensional isentropic fluid flow", Advances inAppL Mech. 3 (1953), pp. 21-59.14. T h e variable ν was introduced by B . R I E M A N N , cit N o t e 11.23, who used theso-called Riemann invariants r = (v + u)/2 = ξ/2 and s — (v — u)/2 = η/2 in placeof u and v. Here £ and η are the characteristic variables of Sec. 12.4, cf.
E q . (10.6).R. LIPSCHITZ ["Beitrag zu der Theorie der Bewegung einer elastischen Flussigkeit",J. reine angew. Math. 100 (1887), pp. 89-120] extended Riemann's discussion, in particular to the case when gravity force acts.15. I t was in discussing Eq. (27') that Riemann developed his theory of integrationof hyperbolic differential equations, see N o t e 11.47.16.
T h e general (p,p)-relationleading to an equation of the type (34) has beengiven by R. SAUER, "Elementare Losungen der Wellengleichung isentropischer Gasstromungen", Z. angew. Math. Mech. 31 (1951), pp. 339-343.17. Equation (34) is a special case of what is now called the Euler-Poisson-Darbouxequation: G. DARBOUX, cit. N o t e 11.47, pp. 54-70. See also L. E U L E R , "Institutionescalculi integralis", Opera Omnia, Ser. 1, Vol. 13, Leipzig and Berlin: Teubner, 1914,pp.
212-230; and S. D . POISSON, " M o m o i r e sur l'intogration des Equations linoairesaux differences partielles", J. ecole polytech. Ser. 1, 19 (1823), pp. 215-248. Recentmathematical interest in the equation and its generalization has been stimulatedmainly by the work of A . W E I N S T E I N , see for example " O n the wave equation and theequation of Euler-Poisson", Proc. Symp. Appl. Math. (A.M.S.)5 (1954), p. 137-147.18.
For either quotient these values are κ = (2N + 3)/(2iV + 1) where Ν is anyinteger. In particular Ν = 1 gives κ = % which is the value of y for a monatomicSas. T h e o r r e s p o n d i n g values of n are then: n = 1 for z = U or pt, n = —1 for VnCHAPTER477IIIArticle 12or χ — ut, η = 2 for ψ, and η = —2 for I. Thus, the physically most interesting casesof monatomic and diatomic gases correspond to mathematically simple equations(34).19.
Equations (37) and (38) give Euler's solution of E q . (34), see N o t e 17. In amore compact notation the equations may be written:z = i^+t ( - -(37)dv/\V(38)Z n(—),)n= (I l Y\v dv/\VZ , = (-l)»zo/(2n -χ» = ( - 1 ) - W ( 2 ml h \\ ν /1l)(2n -3) · • · 1,j- 3) (2m -5) · · · 1.T h e second of these is due to A . E. LOVE and F.
B. PIDDUCK, "Lagrange's ballisticp r o b l e m " , Phil. Trans. Roy. Soc. London, A222 (1922), pp. 167-226. T h e first followsfrom it by a correspondence due to G . DARBOUX, cit. N o t e 17.20. R . v. M I S E S , "One-dimensional adiabatic flow of an inviscidfluid",NAVORDRept. 1719 (1951). T h e basic idea is contained in L. EULER, cit. N o t e 17. Integralrepresentations have been used by Ε . T . COPSON ["On sound waves of finite amplit u d e " , Proc.
Roy. Soc. A 2 1 6 (1953), pp. 539-547] and A . G . M A C K I E ["Contour integral solutions of a class of differential equations", J. Rational Mech. Anal. 4 (1955),pp. 733-750].21. T h e solutions (37) and (38) may also be written in terms of the characteristicvariables ξ and η. Thus (see N o t e 19) with Z = ( - l ) » 2 / 2 » ( 2 n - l ) ( 2 n - 3) · · · 1 =Q+G(v),n'--ft + ^( 3 7 )^Also, with Zo = (-\) ~ 2 z /(2mm(0we fi d after some reductionjlmf-Qt+3)(2m -θ Γ " ft + τ;)-Z n^1+^ c F + ^ J -+5) · · · 1 =d^(ξV++G( ),vv) 'mThese formulas are due to G . DARBOUX, cit.
N o t e 17.22. M o r e generally Eq. (34) reads in characteristic form:dZ2nnΒ&η~~ίθΖηξ +η\θξθζΛ_θη )~φT h e Riemann function is thenΩ=( τ ^ Υ πι +«,ί;=(τχ^Υ\ξ\£ + ν /™ + ™>+ ν /where σ = (ξ — ξι)(η — ηι)/(ξ+ η)(ξι + ηι), F is the hypergeometric function andP the Legendre function; for η = —2 we recapture (46). This result was obtained byB. R I E M A N N , cit. N o t e 11.23. Riemann also discussed the case κ = 1 (isothermal:p/p = c , constant) for which (43) is replaced byn2dV2Θξθη+(dVk[—\θξT h e Riemann function in this case isdV\+ — ) = 0;θη J,k =1- .4c478NOTES A N DADDENDAArticle 12Ω = e ^ >V (2k\/7v),kwhere τ = £ — £1 , ? =+0— νι , and I is the Bessel function (of imaginary argument)0of order zero. T h e precise relation between these two Riemann functions is given inG.
S. S. LUDFORD, " T w o topics in one-dimensional gas d y n a m i c s " , Studies inmatics and MechanicsPresentedto Richardvon Mises,MatheN e w Y o r k : Academic Press,1954, pp. 184-191. This paper also gives other examples. T h e Riemann function forthe "telegraphist's e q u a t i o n " ([2], p. 316):&u+u = 0,can be obtained by noticing that forw = e ' ^ + ^ V i t reduces to one of the above kindwith k — i. Hence we findΩ =Λ(2\/(£ -ϋι)(η -m)),where Jo is the Bessel function of order zero.23. R. v . M I S E S , cit. N o t e 20.
Numerical and graphical methods analogous tothose presented at the end of Sec. 16.7 have been given (respectively) by F. SCHULTZGRUNOW, "Nichtstationare eindimensionale Gasbewegung", Forsch. GebieteIngenieurwesens 13 (1942), pp. 125-134, and R. SAUER, "Characteristikenverfahren fur dieeindimensionale instationare Gasstromung", Ing.-Arch.13 (1942), pp.
78-89.24. W i t h suitable interpretation, Riemann's formula (10.17) still applies to suchcases as this, see G. S. S. LUDFORD, " O n an extension of Riemann's method of integration with applications to one-dimensional gas d y n a m i c s " , Proc. CambridgePhil.Soc. 48 (1952), pp. 499-510, or "Riemann's method of integration: Its extensions withan application", Collectanea Math., 6 (1953), pp. 293-323.25.
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