R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 90
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RIESZ," L ' i n t o g r a l e de Riemann-Liouville et le probleme de C a u c h y " , Acta Math. 81 (1949),pp. 1-223. A shorter paper with almost equal title was presented by the same author atthe Conferences de la reunion Internationale des mathematiciens, Paris, 1937 (proceedings published by Gauthier-Villars, 1939).Some useful particular solutions of the two-dimensional wave equation are mentionede.g. in [31], article on "Small Perturbation T h e o r y " by W .
R. SEARS, pp. 118119 (see N o t e 20). See also our comments on the two-dimensional generalized waveequation (7.45) at the end of A r t . 7.Article 525. ERNST M A C H (1838-1916), using the interferometer method invented by himand his son LUDWIG M A C H [Uber ein Interferenzrefraktometer", Sitzber. Akad.Wiss. Wien ,Abt. I I , 98 (1889), p. 1318] observed projectiles flying at a speed faster thansound: Ε . M A C H and L.
SALCHER, "Photographische Fixierung der durch Projektilein der Luft eingeleiteten V o r g a n g e " , Sitzber. Akad.Wiss. Wien, A b t . I I , 95 (1887), pp.764-780; E. M A C H and L. M A C H , " W e i t e r e ballistisch photographische Versuche",Sitsber. Akad.
Wiss. Wien, A b t . H a , 98 (1889), pp. 1310-1326. Compare also [24], Vol.I I , Chapter X I , "Visualization and Photography of Fluid M o t i o n " , p. 578 ff. Mach'smethod was preceded by the famous schlieren method of A . TOEPLER, Beobachtungennach einer neuen optischei\ Methode, Bonn: Cohen, 1864. T h e excellent monograph onErnst M a c h : H . H E N N I N G , Ernst Mach als Philosoph, Physiker und Psychologe, L e i p zig: A .
BARTH, 1915, contains a bibliography of Mach's papers and books to 1912.Mach gave not only the experimental method and results, but also the essentialtheoretical facts regarding what were later called " M a c h c o n e " , " M a c h a n g l e " ,and " M a c h l i n e s " . L. Prandtl (1907) and T . M e y e r (1908) were probably the firstto use these terms.CHAPTER IIArticle 61. With respect to this article and to part of the next we refer the readerto the monograph [11], which contains a wealth of information—known results complemented by original research and abundant historical material. See also the sameauthor's " V o r t i c i t y and the thermodynamic state in a gas flow", Mem. sci. math.119(1952).CHAPTER469IIArticle 62 .
T h e terms " c i r c u i t " and " c i r c u l a t i o n " go back to W . Thomson, L o r d K e l v i n( 1 8 2 4 - 1 9 0 7 ) : W . THOMSON, " O n vortex m o t i o n " , Trans. Roy. Soc. Edinburgh25(1869),pp. 2 1 7 - 2 6 0 ; also, W . THOMSON and P. G. T A I T , Treatise on Natural Philosophy,London and N e w Y o r k : Cambridge U n i v . Press, which first appeared in 1867. N e w edition in two parts, Part I ( 1 8 7 9 ) , Part I I ( 1 8 8 3 ) ; and both parts, Cambridge UniversityPress, 1912.3 . T h e components of curl q were introduced by d'Alembert, Euler, and Lagrange, and used freely but purely formally in eighteenth century studies onhydrodynamics. Cauchy proved in 1841 that the transformation rules for the components of curl q are the same as those for vector components: A . L. CAUCHY, "Momoire sur les dilatations, les condensations, et les rotations produites par un changement de forme dans un systeme de points materiels", Oeuvres Completes, Ser.
2 , Vol. 12,Paris: Gauthier-Villars 1916, pp. 3 4 3 - 3 7 7 .4. T h e important integral transformation ( 6 ) was actually discovered by K e l v i n( 1 8 5 0 ) as evidenced by a letter from K e l v i n to Stokes. T h e theorem was found independently by Hankel: H . H A N K E L , Zur allgemeinen Theorie der Bewegung der Fliissigkeiten, Gottingen, 1861.5. Interpretations of curl q have been given by Stokes and, in several differentways, by Cauchy, cit. N o t e 3 and N o t e 1.12. See further discussion in TRUESDELL[ 1 1 ] , pp. 5 9 - 6 5 ; see also the presentation in HADAMARD [ 4 ] , p.
74 ff.6. These concepts were introduced by Helmholtz: H . v. HELMHOLTZ " U b e r Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen",reine angew. Math. 5 5 ( 1 8 5 8 ) , pp. 2 5 - 5 5 [translated by P. G. T A I T , " O nintegrals of the hydrodynamical equations which express v o r t e x - m o t i o n " ,Phil.Mag., Ser. 4 , 3 3 ( 1 8 6 7 ) , pp. 4 8 5 - 5 1 2 ] . This paper and the one by L o r d K e l v i n cited inN o t e 2 are the basic papers of the H e l m h o l t z - K e l v i n vortex theory.7. W e note that in this definition v o r t i c i t y is a positive quantity like mass orspeed, while circulation can be positive or negative. Cf. also the detailed presentation in v. M I S E S [ 1 6 ] , Chapters I I and I X .8. Instead of the ad hoc derivation of our text one may compute—in analogy toEq.
( 2 . 2 8 ) — t h e rate of change (d/dt) J (o &*dl where a depends on x,y,z,t and C(t)moves with velocity q. T h e result is (see [ 1 4 ] , p. 4 5 6 )C— + grad (a-q)-q X curl q»dl,and for a = q and C a closed curve— + grad q — q X curl q J »dl =2as in E q . ( 1 1 ) .9 . K e l v i n ' s theorem is given in the paper quoted N o t e 2 .10. Accordingly, one may define an "acceleration potential", φ*, by dq/dt = grad φ*.This was done by Euler ( 1 7 5 5 ) , N o t e 1.3. Obviously by means of φ* some formulasof our text can be rewritten.
T h e fact that the curl of the acceleration vanishes wasknown to d'Alembert ( 1 7 5 2 ) , N o t e 1.4.11. In Sec. 4 the concept of " f i l a m e n t " or of " t u b e of infinitesimal cross s e c t i o n "may be avoided by using a limiting process or by noting that a vortex line is the intersection of any two tubes on which it lies.12. For generalization cf. Sec. 2 4 . 1 .13. Flows for which q X curl q = 0 are called Beltrami fields: E. BELTRAMI, " C o n -470NOTES A N DADDENDAArticle 6siderazioni idrodinamiche", Rend. ist. Lombardo, Ser.
2, 22 (1889), pp. 300-309. M a n yof their properties were previously discovered by I . Gromeka (1881). See TRUESDELL[11], p. 24 ff. and p. 97 ff., for more information and for literature. These flows areinteresting as a link between irrotational flow and general rotational flow.14. I t is known that Helmholtz' original proof of the first vortex theorem is notrigorous. Compare for example LAMB [15], p. 206, and the reference to Stokes there,p.
17. In Sec. 6.6, v. Mises reproduces Helmholtz* original argument—except that heconsiders nonconstant density. Several proofs of the Helmholtz theorems and ofvarious generalizations are presented in TRUESDELL [11]. See also SOMMERFELD [17],p. 130 ff., and N . J . KOTSCHIN, I. A . K I B E L and N . W . ROSE, TheoretischeHydromechanik, Vol. I , Berlin: Akademie Verlag, 1954, p. 138: "Friedmann's T h e o r e m " .Article 715. T h e term " i r r o t a t i o n a l " was introduced by LORD K E L V I N , cit.
N o t e 2.16. This theorem and much of the vortex theory—in the case of an incompressiblefluid—was anticipated in "Cauchy's e q u a t i o n s " [cf. for example LAMB, [15], p. 205,Eq. (3)] which may be generalized to compressible flow.17. T h e exact solution given in this section was probably first considered by G. I .TAYLOR, "Some cases of flow of compressible fluids", ARC Repts.
& Mem. 1382 (1930).A l s o : " R e c e n t work on the flow of compressiblefluids",London Math. Soc. 5 (1930),pp. 224-240. Cf. also H . BATEMAN, " I r r o t a t i o n a l motion of a compressible inviscidfluid", Proc. Natl. Acad. Sc%. U. S. 16 (1930), pp. 816-825.18. This problem is studied in Sec. 17.4 by means of the "hodograph representat i o n " , which will be explained in A r t . 8. See also N o t e 23.19.
This problem—from a different point of view—was considered by G. I .TAYLOR, cit. N o t e 17.20. I t is mentioned for the record that von Mises left an alternative version (complete with drawings) of the end of Sec. 5, and of the whole Sec. 6. T h e method usedthere is closer to that presented here for radial flow, as well as to T a y l o r ' s method.H e introduces new dimensionless variables £ = (2χα /Γ)τ, η = q /a , thereby excluding purely radial flow: Γ = 0, and derives the differential equationrβdr\V2 -I 2 -a(κ - I V + (3 κ)/?(κ + I V - ( - 1)/£ 'κwhich is then discussed: the same results as in the text are derived.21.
Considering flow past a semi-infinite cone, T a y l o r and Maccoll obtained anexact solution for steady irrotational inviscid compressible flow with axial symmetry. G. I . T A Y L O R and J . W . MACCOLL, " T h e air pressure on a cone moving at highspeeds", Proc. Roy. Soc. A139 (1933), pp. 298-311. Compare for example [24], V o l . 1,p. 185 ff. For numerical tables see Z . K O P A L , Tables of Supersonic Flow around Cones,Cambridge, Mass.: Μ . I . T .
Publication, 1947. See also graphs in [35], Sec.III.Article 822. A very important application of these results appears in the "one-dimensional"or " h y d r a u l i c " treatment of channel flow (see Sec. 25.1).23. T h e initial idea of the hodograph representation seems to be due to Helmholtzand to Riemann: H .
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