R. von Mises - Mathematical theory of compressible fluid flow, страница 97

I . W a v e fronts. I I .Waves of moderate a m p l i t u d e " , Quart. J. Mech. Appl. Math. 5 (1952), pp. 257-291.Article 1930. T h e concept of a limit line but not the term appears in G. I . Taylor's previouslyquoted papers, cit. N o t e 11.17, and in the paper of M . U.

CLAUSER and F. H . CLAUSER," N e w methods of solving the equations for the flow of compressible fluids", Thesis,California Inst. Technol., 1937. In 1937, in the paper quoted in N o t e 13, W . Tollmiendefines and discusses an exact transonic solution which exhibits a " G r e n z l i n i e " (limitline) beyond which the flow cannot be continued, and he establishes some of its properties. In 1940 F .

Ringleb, in the paper cit. N o t e 12, gave an example of a limit linewith cusps and studied it in detail. T h e whole situation is discussed and new resultsadded by T . v. K A R M X N , "Compressibility effects in aerodynamics", J.Aeronaut.Sci. 8 (1941), pp. 337-356; see particularly p. 352 ff. (Cf. also the discussion in ourSec. 25.4). In a second paper by F .

RINGLEB (which, however, is open to certainobjections), " U e b e r die Differentialgleichungen einer adiabatischen Gasstromungund den Stromungsstoss", Deut. Math. 6 (1940), pp. 377-384, several results previously found by him and others for special cases are proved for the general case.T h e "Stromungsstoss" is defined as the locus of points where the acceleration becomesinfinite and where the stream lines have cusps. T h e problem is reconsidered by W .TOLLMIEN, "Grenzlinien adiabatischer Potentialstromungen", Z. angew.Math.Mech. 21 (1941), p. 140-152, and new results are added.

None of these earlier Germanpapers, however, discusses in general the cusps of the limit line (see our Sec. 4 ) . Thisholds also for the presentation in COURANT-FRIEDRICHS [21], pp. 62-69 and pp. 256259. A flow similar to Ringleb's was found and discussed by G. TEMPLE and J . Y A R WOOD, "Compressible flow in a convergent-divergent n o z z l e " , A.R.C. Repts. & Mem.2077 (1942).CHAPTER487IVArticle 19T h e physical and mathematical significance of limit lines is considered in the paperH . S. T S I E N , " T h e limiting line in mixed subsonic and supersonic flow of compressiblefluids", Ν AC A Tech. Notes 961 (1944).

Compare also H . S. T S I E N and Υ . H . K u o ,"Two-dimensional irrotational mixed subsonic and supersonic flow of a compressiblefluid and the upper critical Mach number", Ν AC A Tech. Notes 996 (1946). A system­atic exposition containing also a careful discussion of the cusps of the limit line, ofsome singularities of higher order, etc., is given by J.

W . CRAGGS, " T h e breakdownof the hodograph transformation for irrotational compressible fluid flow in two di­mensions", Proc. Cambridge Phil. Soc. 44 (1948), pp. 360-379. F o r further literaturesee Notes 32 and 35, and Sec. 25.4.31. Often the hodograph image of a limit line is likewise called a limit line (andthe same for the branch l i n e ) . This is similar to our use of the word " s t r e a m l i n e " .32. Our method differs from that used by Craggs, Lighthill, [24],Courant and otherscit.

N o t e 30. T h e same method is used in R. E. M E Y E R , "Focusing effects in two-di­mensional supersonic flow", Phil. Trans. Roy. Soc. London A242 (1949), pp. 153-171,a paper which deals not with the basic properties of branch lines and limit lines, butwith more refined problems from the point of view of Riemannian geometry. J.M A N D E L [Squilibrespar tranchesplanesdes solidesa la limited'ecoulement,Paris:Louis Jean, 1942] studies in this way the mapping of the physical plane onto the"stress g r a p h " in the theory of the plane "perfectly plastic b o d y " . Our presentationis decisively influenced by Mandel's.33.

Equations (7), which are adequate for the purposes of this article, may beplified if a ^ 90°. PutA=sin 2a(i-«7Q')'consider Q rather than q as the independent variable, and denote by differentiationwith respect to Q. Then, introducing the function T(Q) by T /T = —2 cos 2a/Aandei , e by hi = ei/y/T,h= e /y/Twe find by a straightforward computation thesimple equations Adei/θη = e , Αθβ /θ£ = e\ \ from these we obtainλy2222dd—2A2θ&η+dei2de—2A A* — — ei = 0θηandA22θξθηde+ AAX.—2-e = 0.2d£34. Geometrically, the situation may also be understood in the following way.T h e correspondence between the x,i/-plane and £,ri-plane defines in four-dimensional£,2/,£,i7-space a two-dimensional subspace, S.

T h e manifold S consists of t w o sheets,connected along the apparent contour, whose projections on the x,i/-plane cover aregion twice. T h e line <£i in the x,?/-plane is part of the projection of the apparentcontour, and similarly for £> · Consider curves through a pointon S which trav­erses the apparent contour. Such curves are projected into curves through Μ ( F i g .126) which are either tangential to the projection <£i or, exceptionally, have cuspsthere. Compare also Hugoniot's ( N o t e I I I . 3 2 ) geometrical ideas [in relation to the(x,£)-problem] which led to the term " e d g e of regression".235.

Most of the definitions and results in this section are new. A few remarksrelating to the subject are in SAUER [29], p. 229, and (partly incorrect) in Ringleb's sec­ond paper, cit. N o t e 30, and, for a somewhat analogous problem, in Mandel's mono­graph, cit. N o t e 32. T w o recent papers are: H .

GEIRINGER, "Grenzlinien der H o d o graphentransformation", Math. Z. 63 (1956), pp. 514-524, and G. S. S. LUDFORD andS. H . SCHOT, " O n sonic limit lines in the hodograph m e t h o d " , ibid. (1957), pp. 229-237.T h e proof of the geometric characterization of the sonic limit line, p. 324 of our textis due to Ludford and Schot. In addition, their paper contains a new example of a488NOTES A N DADDENDAArticle 19sonic limit line. Recently S. H . Schot called our attention to the fact that in a quitenormal example (Sec.

20.3) the derivative φ does not exist at the sonic point of thelimit line £i . Hence, the classification of limit-type singularities based on the behav­iour of ψ^ ,ψηίη addition to that of, φ , which had been proposed in these twopapers, has to be modified and it does not appear in the present book.ξη36. Consider again the two-dimensional manifold S in z,y,£,?7-space, see N o t e 34.T h e original of 6i in four-space belongs to the apparent contour of §. I t separatest w o sheets of S which when projected onto the £,?7 -plane cover a portion of this planetwice.37.

On the original § in four-space, the tangent to such a line is perpendicular t othe£,?7 -plane at its intersection with the apparent contour; hence in the projectiononto the £,*?-plane it shows a cusp. In the projection onto the z,i/-plane, it presentsan inflexion.38. M . J. LIGHTHILL, " T h e hodograph transformation in trans-sonic flow.

I. Sym­metrical channels", P r o c , Roy. Soc. A191 (1947), pp. 323-341. H e defines and discussesbranch lines in two-dimensional transonic flow. See N o t e V.56 regarding Cherry'swork. W e found a branch line—for an (x,t)-problem—inChapter I I I , see Fig. 62.39. Regarding the (x,t)-problem,see P. M . STOCKER and R . E. M E Y E R , " A noteon the correspondence between the z,£-plane and the characteristic plane in a prob­lem of interaction of plane waves of finite a m p l i t u d e " , Proc.

Cambridge Phil. Soc.47 (1951), pp. 518-527.Article 2040. T h e basic source for this article, and in particular for Sees. 1, 2, and 6, isCHAPLYGIN'S paper, cit. N o t e 5.41. Chaplygin, in view of the problem of the subsonic j e t , uses "2n"most authors use "n".where we and42. Regarding literature on the hypergeometric function we mention the presen­tation in WHITTAKER-WATSON [8], the monograph K A M P E DE FERIET [5], and themonograph by F.

K L E I N , Vorlesungeniiber die hypergeometrische Funktion,Berlin:Springer, 1933. See also Notes 49 and 50.43. T h e right side is the sum of two independent solutions of the hypergeometricequation, namely, y = Ayi(r)+ B2/2M, where yi(r) — F(a,b,a + 6 — c + l ; l — τ )and 2/2(7-) = (1 — r) ~ ~ F(c— a,c — b,c + 1 — a — b;l — τ). T h e coefficients arechosen in such a way that y(0) = l,'and?/(l) = F(a b,c;l).("Gauss'transformation".)cahf44. For a discussion of solutions with singularities at an arbitrary point of thesubsonic region ["fundamental solutions" etc.] see S. BERGMAN, "Two-dimensionalsubsonic flows of a compressible fluid and their singularities", Trans.

Am. Math.Soc. 62 (1947), pp. 452-498. Recently R. F I N N and D . GILBARG, " A s y m p t o t i c behaviorand uniqueness of plane subsonic flows", Communs. Pure Appl. Math. 10 (1957),pp. 23-63, showed that the singularities introduced by Bergman are the most generalthat can occur in flows which are subsonic in a neighborhood of infinity.

Fundamentalsolutions for equations of " m i x e d " t y p e (see A r t . 25) have been studied by P. G E R ­MAIN, " R e m a r k s on the theory of partial differential equations of mixed type andapplications to the study of transonic flow", Communs. Pure Appl. Math. 7 (1954),pp. 117-143, see Part I I .45. Following Chaplygin, we use here the variable τ = q /qsince this choiceleads to the hypergeometric equation (7')> whose theory is well known. SeveralGerman authors [F. RINGLEB (cf. N o t e 12), R.

SAUER [29], and G. H A M E L (cf. N o t e9)] use q rather than τ ; this leads to a second-order ordinary differential equation, thesolutions of which are used in a way similar t o that of the hypergeometric functionsin Chaplygin's theory.22mCHAPTER489IVArticle 2046. See F. RINGLEB, cit. N o t e 12. This example is remarkable as one of the firstinstances of smooth transonic potential flow, as well as of an embedded supersonicregion, and an interesting limit line with cusps. (See further comments, Sees. 25.3and 25.4).47. T h e flow (21) has been investigated by G.

TEMPLE and J. YARWOOD, cit. N o t e30. T h e case η = —1 is exceptional, since, with the notations of E q . ( 8 ) , not onlyc_i but also α_ι vanishes. Hence φ-ι is actually indeterminate and may be defined(see [24], p. 233) as:=Φ-Μlimφ (τ).ηn--lT h e result isiMT)= τ""2+ ~,2{κ~it—1)ΦιΜ.Substituting for φι (τ) the expression found in the text we obtain*_,(r) =Δκτ""*-I r-i»(l -IKr) " < ' - ,nwhich is in fact a linear combination of the t w o particular solutions (15) and (21).48. T a k e for example m = 2, or a = π/2.

This solution, which represents compres­sible flow within a corner, was investigated by J. W I L L I A M S , " T h e two-dimensionalirrotational flow of a compressible fluid in the acute region made by two rectilinearw a l l s " , Quart. J. Math., Oxford Ser. 2 (1949), p. 129 ff.

W e have in this case φ =ATF(2.5,— 3,3;r) sin 20, where the hypergeometric function reduces to a polynomialof degree three in r. (See also N o t e 50.)49. Following up earlier work by Gauss and by E. GOURSAT, Sur Vequation differentielle lineaire qui admet pour Vintegrale la s'erie hyper geometrique, Paris, 1881,Lindelof has indicated a solution valid in this case, which involves a logarithmic term:E. L. LINDELOF, "Sur l'int£gration de l i q u a t i o n difforentielle de K u m m e r " , ActaSoc.