R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 88

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I N T E G R A T I O N T H E O R Y A N DSHOCKSthat for the Frankl problem holds, then less than the whole contour issufficient to determine a solution of our problem. Such a conjecture isvery plausible. I t is hard to imagine that the greater complication of thedifferential equation would change an incorrectly set problem into a cor­rectly set one. T h e conclusion would be that smooth potential flow withsupersonic pockets past an arbitrarily given profile of the type defined inSec. 2 does not exist in general.7778I n the light of this conjecture let us once more recall the procedure usedin constructing solutions in the hodograph which were then transformedto the physical plane.

First we determined a branch of a stream functionwhich satisfied a given condition (that it reduces to a given function ψ asq—> oo) and then we computed analytic continuations of this branch.N o w the solutions which have actually been obtained in this way are of anartificial regularity (they are for instance analytic functions in the hyper­bolic region, which is a very unusual situation), so that we avoided typicalfeatures of hyperbolic problems and obtained only certain regular solutions.W e did not solve a boundary-value problem but applied a quite differentmethod of construction of flows, which yielded, for a given Af and P , asmooth transonic flow past a specific contour P , known only a posteriori.Combining these considerations with the experimental evidence, etc., wemight reason as follows.

T h e contours Pconstructed so far by the hodo­graph method are artificial and the corresponding transonic flows can beconsidered as exceptional. T h e set of all contours P(depending on Pand on M°°) obtainable by the hodograph method, if the original P are ofa reasonable generality, has not been identified. However, the contoursΡ Μ (and corresponding flows) resulting in this way will form a fairly re­stricted set and an arbitrarily given Ρ will, in general, not be a Ρ · Also,shapes actually used in experiments may not have the properties necessaryto make them belong to the "exceptional" profiles.0m000MM0M0MI n summarizing the content of Sees. 2 to 6 we must first of all admit thata complete mathematical theory which can be successfully confronted withobservations is lacking; what we have are important but rather isolatedbits of information.

W e know examples of exact solutions—obtained bythe hodograph method—but they are not solutions of the boundary prob­lem in question; and we know of (supposed) numerical solutions of bound­ary-value problems for given contours (Sec. 3 ) . W e saw the agreementbetween these mathematical results and the (mathematical) limit-linestatements of Friedrichs and others (Sec.

4 ) . Our theory does not yielda mathematical counterpart to the observed nonisentropic deceleration(Sec. 3 ) . A hint in the direction of an explanation of these discrepancies issupplied by the proved sensitivity of solutions as seen clearly in a par­ticular instance (Sec. 5 and N o t e 73). And an even stronger suggestion:25.6CONJECTURES ON E X I S T E N C E IN T H ELARGE463If certain recent results are valid under physically relevant conditions, thenthe key for resolving the various difficulties and contradictions is providedby the fact that the boundary-value problem which we took as point ofdeparture is incorrectly set. This, as well as other points discussed in thisarticle must, for the time being, be left undecided.NOTES A N D ADDENDACHAPTER IArticle 11. The principle of the text is NEWTON'S (1642-1727) lex secunda, the second ofthree axioms formulated in the first pages of his Philosophiae NaturalisPrincipiaMathematica.

First, second and third editions: London 1687, 1713, 1726. (Translatedby A . M o t t e , London, 1729; revised by F. Cajory, Berkeley, 1934). T h e three axiomsare at the very beginning, preceded only by the 'definitions". T h e famous secondaxiom reads: " L e x I I . Mutationes motus proportionalem esse motrici impressae etfieri secundam lineam rectam qua vis ilia i m p r i m i t u r . " [Law I I . T h e change of motion is proportional to the motive force impressed, and is made in the direction of theright line in which that force is impressed.]4For today's student of mechanics the three axioms form the starting point. T h e yrepresent at the same time the completion, perfection and generalization ofideas due mainly to Galileo (1564-1642) and Huygens (1629-1695), but lead far beyond the achievements of these great predecessors. An enlightening physical andlogical analysis can be found in E.

MACH'S Mechanik [10]*.2. This rule was used implicitly by L. Euler (1707-1783) and by J. L. Lagrange(1736-1813): L. EULER, "Sectio secunda de principiis motus fluidorum", Novi Commentarii Acad. Set. Petrop. 14(1769), pp. 270-386, and "Principes goneraux du mouvement des fluides", Hist. Acad. Berlin 11 (1755), pp. 274-315, which appeared in 1770and 1757 respectively; J.

L . LAGRANGE, "Momoire sur la thoorie du mouvement desfluides", Oeuvres, Vol. 4, Paris: Gauthier-Villars, 1869, pp. 695-748; the paper appeared first in 1783.For details on this subject see TRUESDELL [11], p. 42, and [13], p. xc.3. L. EULER, cit. N o t e 2. T h e paper in question is the second of three basicpapers which all appeared in 1757 and constitute a treatise on fluid mechanics (see[13], p. L X X X I V f f .

) .In this paper the concept of velocity field is explicitly formulated (though indications appear in earlier works of Euler and others), and in the first and second ofthese three papers the central concept of the pressure field in hydrodynamics isfully explained. (In this connection, see also N o t e 10, regarding J. Bernoulli.) T h eorigin of many of the fundamental ideas developed in this work is found in L. EULER,"Principia motus fluidorum" (completed 1752), Novi Commentarii Acad. Sci.

Petrop.6 (1756-1757), pp. 271-311, which appeared in 1761 but was completed prior to theabove (see [13], p. L X I I ) .4. If the state of motion is given, for all t, for each point (x,y,z) as in Eq. (1), wemay call this the " s p a t i a l " description, while the Lagrangian equations give the' history of each particle, the " m a t e r i a l " description. T h e spatial description was par* We shall refer to the list of reference books by numbers in square brackets.464CHAPTERI465Article 1tially formulated in 1749 by J. L. D'ALEMBERT (1717-1783) in his Essai d'une nouvelle theorie de la resitance des fluides, Paris: David l'aino, 1752, and generalized byE U L E R (1757), cit.

Note 2, who later gave also the material description. Severalauthors, e.g. H. LAMB [15], have pointed out that the usual terminology ofEulerian and Lagrangian equations is unjustified.5. In so far as the continuity equation expresses conservation of mass, its oxiginmay be seen in Newton's lex secunda (see (a), p.

1 in our text). The differential equa­tion itself in various special cases (plane flow, axially symmetric flow) was firstobtained by D'ALEMBERT, cit. Note 4. The general equation for "spatial" as well asfor "material" variables is due to EULER, see the two papers discussed in Note 3.(Compare also TRUESDELL [11], p. 50, on the d'Alembert-Euler continuity equation.)6. The author's presentation, based on equations ( I ) , ( I I ) , and ( I I I ) , as developedin Arts. 1-3, differs from that in most modern textbooks where the physical pointof view is emphasized. See for example L.

HOWARTH [24], Vol. 1, Chapters I and I I .For the author's point of view compare also R. v. MISES, "On some topics in the funda­mentals of fluid flow theory", Proc. First Natl. Congr. Appl. Mech., Chicago (1950),pp. 667-671.7. The theory based on Eq. (5c) is dealt with in our book in Sees.

17.5 and 17.6.8. The equation of state for a perfect gas in equilibrium, connected with the namesof Boyle (1660), Mariotte, Amontons, Gay Lussac, and Charles, has been widelyknown since 1800. In precisely the modern form, it was used freely by Euler, but didnot appear again in the hydrodynamical literature until used by Kirchhoff (18241887).In some presentations no distinction is made between the terms "perfect" and"ideal". In our book, the term " i d e a l " is used for "inviscid and nonconducting".The term "perfect" is defined in Eq.

(1.6). Incidentally, the term "elastic" introducedon p. 7 is of very long standing in fluid dynamical literature.9. In this work, the term "isentropic" is used if the entropy is the same every­where and at all times. The term "strictly adiabatic" (see p. 9) applies to the casewhere the entropy is constant for each particle but varies from particle to particle.To assist the reader, we mention that L. HOWARTH [24] calls this latter case "isen­tropic" and uses "homentropic" for the case of entropy constant throughout.Article 210. Equation (2.20) which we derived here from Newton's equation (1.1) (comparealso p. 18) is generally attributed to Daniel Bernoulli (1700-1782): D. BERNOULLI,Hydrodynamica,sive de viribus et ynotibus fluidorum commentarii,Strassburg, 1738(hence, some years before Euler's general equation, cit.

Note 3). Much credit is duealso to his father John Bernoulli (1667-1748): JOHN BERNOULLI, "Hydraulica nuncprimum detecta ac demonstrata directe ex fundamentis pure mechanicis, Anno 1732",Opera Omnia, Vol. 4, Lausanne and Geneva: Μ. M. Bousquet, 1742, pp. 387-493. Theequation for steady flow of an incompressible fluid, discovered but imperfectly de­rived by 1).

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