R. von Mises - Mathematical theory of compressible fluid flow, страница 102

10 (1952), pp. 177-184; C. S. MORAWETZ and I . I . KOLODNER, " O n the non­existence of limiting lines in transonic flows", Communs. Pure Appl. Math. 6 (1953),pp. 97-102.71. See J . HADAMARD, cit. N o t e 1.24, p. 4. This definition is not explicitly givenon page 4 or elsewhere in his book.

However, it follows clearly from the lucid dis­cussion in his Chapters I and I I .72. A n alternative proof, a little longer but closer to v. Mises' ideas and perhapsmore direct than the one given in our text, would be as follows. W e prove as before the"monotonicity l a w " . Then, following A . A . N I K O L S K I I and G.

I. TAGANOV, cit. N o t e69, we prove that θ and q are monotonic functions along a given segment of a charac­teristic, say a C , which is inside the supersonic pocket, with the C~ originating atthat C ending at the sonic line. N e x t , consider a point Ρ on the contour in thesupersonic region, and the C , C~ at Ρ both in the direction towards the sonic lineS; call ds , dsi the respective line elements. F r o m the monotonic change of q alongt h e C and C~ it follows that both dq/dsi and dq/ds are ^ 0.

Also by a brief computa­tion+++2+21dq1 dqq sin a ds=2k +q dsΛcot a,and1dq,1 dq—:= k —cot a,q sin a dsiq dswhere k is the curvature at P. If then k = 0 along an arbitrarily small piece of thecontour, it follows that there also dq/ds = dq/dsi = dq/ds = 0, and the conclusionsp. 458 line 17 ff. apply.73.

In an unpublished N o t e " N o n - e x i s t e n c e of transonic flow past a profile with2500NOTES A N DADDENDAArticle 25vanishing curvature" C. S. Morawetz has proved a stronger result than that of A . A .Nikolskii and G. I. Taganov. Suppose 1) that the curvature, fc, of any streamline is acontinuous function of its length in the supersonic region, including the profile itself;2) that k does not vanish at either of the two sonic points on the profile. If then kvanishes at a point on the profile in the supersonic region then either θφ/dq or θφ/θθmust become infinite at at least one point.74. F.

I . F R A N K L , ' O n the formation of shock waves in subsonic flows with localsupersonic v e l o c i t i e s " , Prikl. Mat. Meh. 11 (1947), pp. 199-202 [translation: Ν AC ATech. Mem. 1261 (1950)]; G. GUDERLEY ' O n the presence of shocks in mixed subsonicsupersonic flow p a t t e r n s " , Advances in Appl. Mech. 3 (1953), pp. 145-184, and hisforthcoming book Theorie schallnaher Strdmungen, Berlin: Springer-Verlag, 1957;A .

BUSEMANN, " T h e drag problem at high supersonic speeds", J. Aeronaut. Sci. 16(1949), pp. 337-344, and " T h e nonexistence of transonic flows", Proc. Symp.Appl.Math. (A. M. S.) 4 (1953), pp. 29-39. T h e highly suggestive arguments of Busemannand Guderley concern the focusing of disturbances at a sonic point of the profile,which may cause a considerable change in the flow pattern. A .

R . M A N W E L L [ " T h evariation of compressible flows", Quart. J. Mech. Appl. Math. 7 (1954), pp. 40-50]applies the perturbation theory to transonic vortex flow, that is, a circulatory flowoutside a circular cylinder (Sec. 17.4). H e shows that to small changes of the boundarycorrespond in general small changes of the flow pattern. H o w e v e r , at certain discretespeeds a kind of resonance phenomenon arises.75.

Some of the numerous investigations on the Tricomi equation are quoted inL. BERS, cit. N o t e 60, and in F. G. TRICOMI, cit. N o t e 11.27.76. See F. FRANKL, cit. N o t e 74. Another generalization of Tricomi's problem hasbeen considered by Gellerstedt. Values of u are given on C o , and in addition onthe characteristics O n , O r (Fig. 174) or on Sin and S2T2 .

S. GELLERSTEDT, "Quelques problemes mixtes pour l i q u a t i o n τ/™ζ χ + z = 0 " , Ark. Mat. Astron. Fys. 26A(1938), N o . 3. In a recent paper Μ . H . PROTTER ["Uniqueness theorems for theTricomi problem. I I " , J. Rational Mech. Anal. 4 (1955), pp. 721-732] summarizesprevious results and proves a new uniqueness theorem. Compare the preceding paperof this title, ibid.

2 (1953), pp. 107-114, and " A n existence theorem for the general­ized Tricomi p r o b l e m " , Duke Math. J. 21 (1954), pp. 1-7. See also C. MORAWETZ," A uniqueness theorem for Frankl's problem", Communs. Pure Appl. Math. 7 (1954),pp. 697-703.77. T h e nonlinearity of the equation in the physical plane should not decisivelyinfluence the uniqueness problem.

I t can be shown that the difference oi(x,y)of twosolutions of the planar equation az+ 2bz-f cz+ d = 0, where a,b,c,d arefunctions of x,y,z , z but not of z, satisfies a linear equation of the formαω -f 2bo) + Cd3 + do>x + βω = 0. ( A simple proof may be found in D . GILBARG,cit. N o t e 61, p. 235 ff. Cf.

also L. BERS, cit. N o t e 60, p. 98.)78. In his repeatedly quoted article, L. Bers formulates several "conjectured non­existence theorems". Conjecture A relates to nonexistence essentially as explainedin the text. Assume that there exists a transonic flow with a given speed q°°, past aprofile P ; then there exists no smooth potential flow with the same q°° past a profileΡ which differs from Ρ only along a "critical a r c " ( 7 \ T in Fig. 175). Conjecture Ccontains the weaker statement that for the transonic flow past Ρ the "perturbationp r o b l e m " in the classical sense is not well set.

Conjecture Β considers the variationof q .In a recent paper C. S. MORAWETZ ["On the non-existence of continuous transonicflows past profiles, I " , Communs. Pure Appl. Math. 9 (1956), pp. 45-68] has publisheda proof of conjecture C. In the continuation [C. S. MORAWETZ, " O n the non-existenceof continuous transonic flows past profiles, I I " , ibid. 10 (1957), pp.

107-131] she makes2yyΧxxxχχxyxyyyyyyν2xCHAPTERV501Article 25an essential contribution to the basic problem A . Such mathematical results in thisdifficult domain are, at any rate, of great interest. However, only a detailed studyof the implications of the assumptions made and of the results obtained can show towhat extent the results elucidate the situation considered in our text.SELECTED REFERENCE B O O K ST h e asterisk* denotes works that contain extensive bibliographical dataMATHEMATICS1.* Bergman, S., and Schiffer, M . , Kerneltions in Mathematical2. Courant, R.,Physics,Functionsand EllipticDifferentialEqua­N e w Y o r k : Academic Press, 1953.and H u b e r t , D . , Methodender mathematischenPhysik,Vol.II,Berlin: J.

Springer, 1937. Reprinted 1953, N e w Y o r k : Interscience.3. Frank, P. and Mises, R . v . (editors), Die DifferentialundIntegralgleichungender Mechanik und Physik, V o l . I , Braunschweig: F. Vieweg und Sohn, 1930.Reprinted 1943, N e w Y o r k : M a r y Rosenberg.4. Hadamard, J., Legons sur la propagationdes ondes et les equationsdeVhydrodyna-mique, Paris: A . Herman, 1903. Reprinted 1949, N e w Y o r k : Chelsea.5. Kampe* de Foriet, J., " L a fonction hypergoomotrique", Mem. sci.

math. 85 (1937).6. L e v i - C i v i t a , T . , Characteristiquesdes systhmesdifferentielset propagationdesondes, Paris: Alcan, 1932.7. Sauer, R.,Anfangswertproblemebei partiellenDifferentialgleichungen,Berlin:Springer, 1952.8. Whittaker, Ε. T . and Watson, G. N . , A Course of Modern Analysis, 4th ed., Lon­don: Cambridge Univ. Press, 1927. Reprinted 1944, N e w Y o r k : Macmillan.HISTORICAL STUDIES9.* Ka>man, T .

v . , Aerodynamics,SelectedTopicsin the Lightof theirHistoricalDevelopment, Ithaca: Cornell Univ. Press, 1954.10. Mach, E., The Science of Mechanics (transl. b y T . J. M c C o r m a c k ) , 5th ed., Lon­don: Open Court, 1942.11.* Truesdell, C , The Kinematics of Vorticity, Bloomington: Indiana Univ. Press,1954.12.* Truesdell, C , " T h e Mechanical Foundations of Elasticity and Fluid D y n a m i c s " ,J. Rational Mech.

Anal. 1 (1952), pp. 125-171, 173-300.13.* Truesdell, C , Editor's Introduction to LeonhardiEuleriOperaOmnia,Ser. I I ,Vols. X I I and X I I I , Zurich: Orell Fussli, 1954 and 1955.HYDRODYNAMICS14. Frank, P. and Mises, R. v . (editors), Differentialund Integralgleichungen derMechanik und Physik, V o l . I I , Braunschweig: F. Vieweg, 1935. Reprinted1943, N e w Y o r k : M a r y Rosenberg.15.* L a m b , H . ; Hydrodynamics,6th ed., London, Cambridge U n i v . Press, 1932.

R e ­printed 1945, N e w Y o r k : D o v e r .16. Mises, R . v., Theory of Flight, N e w Y o r k : M c G r a w - H i l l , 1945. Reprinted 1958,N e w York: Dover.17. Sommerfeld, Α . , Mechanics of Deformable Bodies (transl. b y G. K u e r t i ) , N e wY o r k : Academic Press, 1950.502503NOTES A N D A D D E N D AAERODYNAMICS18. Ackeret, J., " G a s d y n a m i k " , Handbuchder Physik,1927, pp.

289-342. Translated as Brit. MinistryPubis.V o l . 7, Berlin: J . Springer,of Supply,Repts.andTech.2119.19. Busemann, Α . , " G a s d y n a m i k " , Handbuch der Experimentalphysik,Vol. 4.1, Leip­zig: Akademische Verlagsges., 1931, pp. 341-460. Translated as Brit.Ministryof Supply,2207 through 2212.Repts. and Tech.

Pubis.20.* Carrier, G. F. ( e d i t o r ) , Foundationsof HighSpeed Aerodynamics,New York:D o v e r , 1951.21.*Courant, R., and Friedrichs, K . O., SupersonicFlow and Shock Waves, N e w Y o r k :Interscience, 1948.22.* D r y den, H . L., Murnagham, F. P., and Bateman, H., Hydrodynamics,D o v e r Publications, 1956. Reprint of Natl.ResearchCouncilNew York:U.

S. Bull. 84(1937).23. Durand, W . F. ( e d i t o r ) , AerodynamicTheory, 6 vols, (in particular V o l . 3, D i v .H ) , Berlin: J. Springer, 1934. Reprinted 1943, California Inst, of Technology.24. H o w a r t h , L. ( e d i t o r ) , Modern Developmentsin Fluid Dynamics;High SpeedFlow,2 vols., London and N e w Y o r k : Oxford U n i v . Press, 1953.25. Mises, R .

v., and Friedrichs, K . , Fluid Dynamics(Lecture N o t e s ) , Brown Univ.Summer Session for Advanced Instruction and Research in Mechanics, P r o v i ­dence, Rhode Island, 1941.26. Mises, R . v . , Noteson MathematicalTheoryof CompressibleFluidFlow,Cam­bridge, Massachusetts: H a r v a r d U n i v . Grad. School of Eng., Special Publ.

2,1949.27.* Oswatitsch, K . , Gas Dynamics(English version by G. K u e r t i ) , N e w Y o r k :Academic Press 1957.28. Rayleigh, J. W . S., The Theory of Sound, 2 vols., 2nd ed., N e w Y o r k : D o v e r , 1945.(Historical introduction b y R . B . Lindsay.)29. Sauer, R., Scoulements des fluides compressibles, Paris: Librairie Polytechnique,1951. A l s o : Einfuhrungin die theoretischeGasdynamik,Springer, 1951.30.* Sauer, R., " G a s D y n a m i c s " , Fiat Rev. GermanM a t h . ) , 5, pp. 126-162.31.*2nd ed., Berlin: J.Sci. 1989-1946, Part I I I , (Appl.Sears, W . R . ( e d i t o r ) , General Theory of High Speed Aerodynamics,Princeton,N e w Jersey: Princeton U n i v .