R. von Mises - Mathematical theory of compressible fluid flow (798534), страница 91
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v . HELMHOLTZ, " O n discontinuous movements of fluids", Phil.Mag., Ser. 4, 36 (1868), pp. 337-346 [original in Monatsber. preuss. Akad. Wiss. Berlin125 (1868) pp. 215-228]. Helmholtz used it t o solve problems involving free boundaries(cf. Sec. 20.6). T h e importance of the transformation as a method for obtaining linearCHAPTER471IIArticle 8equations was perhaps first recognized by G. B .
Riemann (1826-1866) in a paper whichis basic for the subject of this book: G. B . R I E M A N N , " U b e r die Fortpflanzung ebenerLuftwellen von endlicher Schwingungsweite", Abhandl. Ges. Wiss. Gottingen,Math.physik. KI. 8 (1858/9), pp. 43-65, or Gesammelte mathematische Werke, 2nd ed., 1892,p. 157 ff.
(reprinted 1953, N e w Y o r k : D o v e r ) . Compare also N o t e I V . 5 .24. L. PRANDTL and A . BUSEMANN, "Naherungsverfahren zur zeichnerischenErmittlung von ebenen Stromungen mit Uberschallgeschwindigkeit", Stodola Festschrift (1929), pp. 499-509. Reprinted in [20], pp. 120-130.25. This is one of the oldest results on compressible fluid flow, due to B . DE S T .V E N A N T and L. WANTZEL, " M o m o i r e et experiences sur l'ecoulement de Fair determine* par des differences de pressions considerables", J.
ecole polytech., Ser. 1, 27(1839), pp. 85-122.26. We refer the reader to the tables [34], [36], [37]. Extensive diagrams are givenin [35]. A textbook comparatively rich in tables is A . FERRI, Elements of Aerodynamicsof Supersonic Flow, N e w Y o r k : Macmillan, 1949.Article 927. Regarding terminology: Equations of the form (2) are called planar by v. Miseswhether or not the coefficients depend on Φ itself. T h e equation for the potential,the stream function, in Arts.
16 and 24, and the particle function, in Arts. 12 and 15,are all of this form. This type of equation is sometimes termed pseudolinear,e.g.in [7]. If the coefficients A, B, C, F depend on x, y, ΘΦ/θχ, ΘΦ/dy, but not on Φ itself, the equation is often called quasilinear.However, in the monograph: F . G.TRICOMI, Lezioni sulle equazioni a derivate parziali, T o r i n o : Editrice Gheroni, 1954,Eq.
(2) is called quasilinear if A, B, C depend on x, y only, while F may depend onx, y, Φ, ΘΦ/θχ, ΘΦ/dy; this case is usually called semilinear. T h e linear equation ofsecond order is always assumed to be of the formAΘ*Φθχ2where Α, Β,...\- Ββ*ΦdxdyΘΦΘΦ+ C — + D +EdyθχΘΦ— + FΦ + G = 0,dy22G depend on x, y only.28. In earlier treatises, characteristics were mainly studied for one partial differential equation of first order, or one partial differential equation of second order, seefor example [3] and A .
SOMMERFELD, Partial differential equations in Physics (trans,by E. G. Strauss), N e w Y o r k : Academic Press, 1949. In contrast to this, in the present text and in [2], [4], [6], [21] systems of equations of first order are in the foreground.This approach was largely developed by J . HADAMARD [4] and by T . L E V I - C I V I T A [6].T h e important book, J . HADAMARD, cit.
N o t e 1.24, which is, in general, mathematically too advanced for our purpose, contains in the Preface and in Chapter I interesting information regarding the earlier literature. A presentation such as the onein our test, with emphasis on "discontinuous solutions" in compressible fluid flow,is given in a very condensed form in the paper quoted N o t e 1.6.29. A detailed discussion of the case η = 2, k = 2 follows in A r t .
10. T h e caseη = 3, k = 2 is considered e.g. in [7], pp. 147-152, and the case η = 2, k = 3 in [29],p. 170 ff.30. If the equation of such a characteristic surface is written as/(xi, £2,· · ·, x ) =constant, then clearly / must satisfy the first-order partial differential equationwhich is obtained from (10') on replacing X„by θ}/θχ(μ = 1, 2,· · ·, η). T h e so-calledMonge cone (see [2], p. 63) of this first-order equation is the cone enveloped byplanes whose normal direction satisfies (10'), i.e. it is the conjugate of the coneformed by the normal directions. (See N o t e 32.)nμ472NOTES A N DADDENDAArticle 931. v. M I S E S , cit.
N o t e 1.6, introduced the concept of a discontinuous solution, inrelation to the distinction of our text between "compatibility relations" and " a d d i tional equations". A set of functions ui , u , · · · , Uk is called a discontinuous solutionof Eqs. (9.6) across a surface S* if ( i ) on both sides of S* all differential equations aresatisfied, and if (ii) at least one of the Ui or its derivatives has a jump across S*.We shall see that in the general problem of Sec. 6 such discontinuities across thecharacteristic surfaces S* actually arise, including " a b s o l u t e " discontinuities, i.e.discontinuities of the variables Ui .2T h e definition leads to a criterion regarding the important question as to whichof the k variables may jump across S*. Obviously, a variable may jump withoutviolating conditions ( i ) , (ii) if its derivative normal to S* does not appear at all in the(k — r) additional equations (see N o t e 38).
Instead of this, both Hadamard andL e v i - C i v i t a use special physical reasoning to show why, in compressible flow, forexample the pressure must not jump and certain velocity components cannot changeabruptly.32. E q . (20) for the three-dimensional steady potential equation may be written(q2x-a )p22+(q2v-a )q22+(?* 2a ) + 2q q„pq + 2q q q2xyz-2q q pzx= 0,where p,q are first-order derivatives of the characteristic surface, and p:q: — 1 =λι:λ2ΐλ3 .
T h e Monge cone of this equation, considered as the partial differential equation of the characteristic surface, is identical with our M a c h cone.33. T h e theory for a system of equations of second order from which the resultsregarding one equation of second order are then immediate, can be found in [4] orin [6], p.
9 ff.34. T h e present discussion applies to the very general "ideal fluid m o t i o n " of Sec.3.6. T h e solution in the case of an elastic fluid where η = k = 4, can be found in [6],p. 63.35. I n Sec. 4 of the paper cit. N o t e 1.6, v . Mises also investigates the characteristics when viscosity and heat conduction are admitted. Under the usual assumptionsthe system then consists of four differential equations of second and one of firstorder. T h e corresponding equation for λ , which is of degree nine, resolves into theproduct of the factor q\i + λ* and of a polynomial of degree eight which can vanish onlyif all λ» = 0.
Hence the only discontinuities in this problem are those related to thefactor q\\ + λ4 . B y making use of prior work of P. Duhem, the same results withoutheat conduction were obtained by G. LAMPARIELLO, "Sull impossibilita di propagazioni ondose nel fluidi viscosi", Atti. accad. nazl. Lincei, Rend. Classe set. fis. mat. eπα*., Ser. 6, 13 (1931), pp. 688-691.36. T h e intersection of an exceptional plane in χ,ι/,ζ-space, for which (27) holdsfor λ , with the x,i/-plane is not in general a Mach line.
Analogously, the intersectionof an exceptional plane in x,i/,2,i-space,with (26) for λ , with the 2,1/,2-space is in general not tangent to the Mach cone.37. J . HADAMARD [4] introduced a different approach, much used in recent workon continuum mechanics (see also the clear and condensed presentation in [6]). Consider a discontinuity surface S, varying in time. If the discontinuity is fixed with respect to the medium, it is called a material discontinuity[example: q\i + λ4 = 0 inEq. (26)]; otherwise it is called a wave (onde) [second factor of Eq. (26)].In this theory the characteristic condition (10) is obtained by a method quitedifferent from that explained in our text.38.
Thus with respect to the triple root, our five original equations are transformedinto three "compatibility relations", plus two "additional equations". These lastare: one, the component of Newton's equation in the λ-direction; t w o , the continuityCHAPTER473IIArticle 9equation. T h e first of these includes the derivative of ρ in the λ-direction; hence thepressure,p, cannot change abruptlyacross a surfaceconsistingof particlelines.Combining this with an analogous consideration of the continuity equation we find thatonly p and the tangential component of q may jump. These important facts are obtainedby v .
Mises, cit. N o t e 1.6, from the principle explained at the end of N o t e 31, whichobviates introducing any separate physical considerations to explain why, for example, ρ may jump and ρ must not.Jumps in these variables, viz., density and tangential velocity, across particlelines (streamlines in the steady case) do in fact occur; they are known as contactdiscontinuities, vortex sheets, etc. (see Sec.
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