Abstract
Newton’s problem of constructing an axisymmetric forebody part of minimum drag is considered. The solution to this problem, although without any explanation, was given by Newton himself in the main work of his life, Philosophiae Naturalis Principia Mathematica. However, Newton’s prefabricated solution was not understood by aerodynamicists who turned to solving Newton’s problem and some of its generalizations in the middle of the twentieth century. A.N. Krylov translated Newton’s Principia into Russian, giving detailed explanations of many of Newton’s statements, including the discussed problem. Moreover, having explained one of these statements, Krylov formulated the necessary condition for the drag minimum, missed by all Newton readers, not just aerodynamicists, but also such an authority on the variational calculus as Legendre. However, even Krylov’s explanations did not help to understand Newton’s solution to the only who had access to them—Soviet aerodynamicists. The main goal of this article is to describe the history of the Newton problem solution, in which the author happened to participate.
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References
A. L. Gonor and G. G. Chernyi, “Bodies of Minimum Drag at High Supersonic Velocities,” Izv. Akad. Nauk SSSR, Otd. tekh. nauk, No. 7, 89–93 (1957) [in Russian].
A. N. Kraiko, Theoretical Gas Dynamics: Classics and Topics (Torus Press, Moscow, 2010), 440 p. [in Russian].
I. Newton, Mathematical Principles of Natural Philosophy (Philsophiae Naturalis Principia Mathematica) / Translated from the Latin by E. Motta, 1793; revised ed. by F. Cajori (University of California Press, Berkeley, CA, 1947), 333–334, 657–661.
A. J. Eggers, Jr., M. M. Resnikoff, and D. H. Dennis, Bodies of Revolution Having Minimum Drag at High Supersonic Airspeeds, Report No. 1306 (NACA, 1957).
G. G. Chernyi and A. L. Gonor, “Nonslender Shapes of Minimum Pressure Drag,” in: Theory of Optimum Aerodynamic Shapes / Ed. by A. Miele (Acad. Press, N.Y.; L., 1965), 373–385.
A. N. Kraiko, “The Determination of Minimal Drag Bodies by Newton’ s and Buzemann’ s Drag Laws,” J. Appl. Math. Mech. 27 (3), 723 (1963).
A. L. Gonor and A. N. Kraiko, “Some Results of the Optimal Forms Study for Super- and Hypersonic Velocities,” in: Theory of Optimum Aerodynamic Shapes / Transl. from Engl. ed. by A. L. Gonor (Mir, Moscow, 1969), 507 p., 455–492 [in Russian].
A. N. Kraiko, Variational Problems of Gas Dynamics (Nauka, Moscow, 1979), 447 p. [in Russian].
Yu. D. Shmyglevskii, Analytical Investigation of Fluid Dynamics (Editorial URSS, Moscow, 1999), 231 p. [in Russian].
A. N. Kraĭko, D. E. Pudovikov, K. S. P’yankov, and N. I. Tillyaeva, “Axisymmetric Nose Shapes of Specified Aspect Ratio, Optimum or Close to Optimum with Respect to Wave Drag,” J. Appl. Math. Mech. 67 (5), 703 (2003).
A. Eggers, “Minimum Wave Drag Non-Slender Bodies of Revolution,” in: Theory of Optimum Aerodynamic Shapes / Transl. from Engl. ed. by A. L. Gonor (Mir, Moscow, 1969), 507 p., 260–274 [in Russian].
A. N. Kraiko, “The Front End of a Given Volume Having Optimum Pressure Drag in the Approximation of Newton’s Law of Resistance,” J. Appl. Math. Mech. 55 (3), 310 (1991).
N. L. Yefremov, A. N. Kraiko, and K. S. P’yankov, “The Axisymmetric Nose Shape of Minimum Wave Drag for Given Size and Volume,” J. Appl. Math. Mech. 69 (5), 649 (2005).
N. L. Yefremov, A. N. Kraiko, K. S. P’yankov, and S. A. Takovitskii, “The Construction of a Nose Shape of Minimum Drag for Specified External Dimensions and Volume Using Euler Equations,” J. Appl. Math. Mech. 70 (6), 912 (2006).
Fluid mechanics. Selected / Ed. by A. N. Kraiko, at al. (Phyzmatlit, Moscow, 2003), 752 p. [in Russian].
G. G. Chernyi, A. L. Gonor, and E. L. Ivanova, Ideal Gas Flows around Bodies at High Supersonic Velocity / Tecnical Report, No. 2794 (TsIAM, Moscow, 1956), 36 p. [see also: A. N. Kraiko, at al. (ed.), Fluid mechanics. Selected, (Phyzmatlit, Moscow, 2003), 37-52] [in Russian].
I. Newton, Mathematical Principles of Natural Philosophy / Translated from the Latin with a commentary by A. N. Krylov (Nauka, Moscow, 1989), 688 p. [in Russian].
A. N. Kraiko, “Newton’s Problem on the Head Part of the Minimum Drag with A. N. Krylov’s Commentaries and Continuation of its Solving History in the XX and Early XXI Century,” in: High-Speed Hydrodynamics and Shipbuilding” dedicated to the 155th anniversary of academician A. N. Krylov / Proc. Intern. Scient. School-Conf. (Chuwash. State University, Cheboksary, 2018), 268 p., 47–56 [in Russian].
Acknowledgments
The author is grateful to A.G. Terent’ev, for it is only thanks to his persistence that this article was created and published originally [18] in the proceedings of the conference "High-Speed Hydrodynamics and Ship Construction" (Cheboksary June 24–29, 2018), dedicated to the 155th birthday of Academician A.N. Krylov.
Funding
The work is supported by the Russian Foundation for Basic Research, project no. 17-01-00126.
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Russian Text © The Author(s), 2019, published in Prikladnaya Matematika i Mekhanika, 2019, Vol. 83, No. 5–6, pp. 734–748.
For the anniversary of a major specialist in the field of gas dynamics and aerodynamics, the editorial board of the journal publishes a memoir article by A.N. Kraiko.
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Kraiko, A.N. Newton’s Problem of the Optimal Forebody: History of the Solution. Fluid Dyn 54, 1009–1019 (2019). https://doi.org/10.1134/S0015462819080056
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DOI: https://doi.org/10.1134/S0015462819080056