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Ideal Compressible Fluid

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Book cover Variational Principles of Continuum Mechanics

Part of the book series: Interaction of Mechanics and Mathematics ((IMM))

Abstract

Ideal compressible fluid can be considered as an elastic body the internal energy of which depends only on the mass density of the body. Therefore, the variational principles formulated for elastic bodies are valid for compressible fluids as well. However, they deserve special consideration because, due to a simplified energy structure, they are enriched by new interesting features.

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References

  1. H. Bateman. Partial differential equations of mathematical physics. Cambridge Univ. Press, Cambridge, 1959.

    MATH  Google Scholar 

  2. V.L. Berdichevsky. Variational principles of continuum mechanics. Nauka, Moscow, 1983.

    Google Scholar 

  3. F.P. Bretherton. A note on Hamilton’s principle for perfect fluids. Journal of Fluid Mechanics, 44:19–31, 1970.

    Article  MATH  Google Scholar 

  4. P. Casal. Principles variationnels en fluide compressible et en magnetodynamique. Rev. Mec. Gren. Anyl., XI:40, 1965.

    Google Scholar 

  5. B. Davydov. Variational principle and canonic equations for ideal fluid. Soviet Physics – Doklady, 69(1), 1949.

    Google Scholar 

  6. I. Deval. Sur la dynamique des fluids perfaits et le principles d’Hamilton. Bulletin de l’Académié de Belgique (Classe des Sciences), t.37:386–390, 1951.

    Google Scholar 

  7. Th. de Donder and F.H. Van der Dungen. Sur les principles variationnelles des milieux continus. Bulletin de l’Académié de Belgique (Classe des Sciences), 35:841–846, 1949.

    MATH  Google Scholar 

  8. S. Drobot and A. Rybarski. A variational principle of hydrodynamics. Archive of Rational Mechanics and Analysis, 11:393–401, 1958.

    Article  MathSciNet  Google Scholar 

  9. V.T. Dubasov. Application of variational methods to solution of aerodynamic problems. Trudi MATI, 42:5–30, 1959.

    Google Scholar 

  10. E. Eckart. Variational principles of hydrodynamics. Physics of Fluids, 3:421–427, 1960.

    Article  MATH  MathSciNet  Google Scholar 

  11. H. Ertel. Uber ein allgeneines Variationprinzip der Hydrodynamik. Abhandlungen preussiche Akademie der Wissenschaften Physical Mathematics Klasse, 7:S:30–41, 1939.

    Google Scholar 

  12. R. Hargreaves. A pressure-integral as kinetic potential. Philosophical Magazine, 1908.

    Google Scholar 

  13. J.W. Herivel. The derivation of equations of motion of an ideal fluid by Hamilton’s principle. Proceedings of the Cambridge Philosophical Society, 51:344–349, 1955.

    Article  MATH  MathSciNet  Google Scholar 

  14. E. Holder. Klassische und relativistische Gasdynamik Variations-problem. Mathematical Nachrichten, Band 4:366–371, 1950.

    MathSciNet  Google Scholar 

  15. H. Ito. Variational principle in hydrodynamics. Progress in Theoretical Physics, 9(2):117–131, 1953.

    Article  MATH  Google Scholar 

  16. H. Jeffreys. What is Hamilton principle? Quarterly Journal of Mechanics and Applied Mathematics, 7:335–337, 1954.

    Article  MATH  MathSciNet  Google Scholar 

  17. A.N. Kraiko. Variational principles for perfect gas flows with strong discontinuities expressed in Eulerian variables. Journal of Applied Mathematics and Mechanics (PMM), 45(2):184–191, 1981.

    Article  MATH  Google Scholar 

  18. B.G. Kuznetsov. On the second Bateman variational principle. Proceedings of Tomsk U., 144:117–124, 1959.

    Google Scholar 

  19. C.M. Leech. Hamilton’s principle applied to fluid mechanics. The Quarterly Journal of Mechanics and Applied Mathematics, XXX(Pt. I):107–130, 1977.

    Article  MathSciNet  Google Scholar 

  20. L. Lichtenstein. Grundlagen der Hydrodynamik. Springer-Verlag, Berlin, 1929.

    Google Scholar 

  21. C.C. Lin. A new variational principle for isenergetic flow. Quarterly Applied Mathematics, 9(4):421–423, 1952.

    MATH  Google Scholar 

  22. C.C. Lin. Liquid helium. In International school of physics course, volume XXI, Academic Publisher, New York, 1963.

    Google Scholar 

  23. C.C. Lin and S.L Rubinov. On the flow behind curved shocks. Journal of Mathematical Physics, 27(2):105 – 129, 1949.

    MathSciNet  Google Scholar 

  24. M.V. Lurie. Application of variational principle for studying discontinuities in continuum media. Journal of Applied Mathematics and Mechanics (PMM), 30(4):855–869, 1966.

    Article  Google Scholar 

  25. M.V. Lurie. Application of variational principle for studying the propagation of discontinuities in continuum media. Journal of Applied Mathematics and Mechanics (PMM), 33(4):586–592, 1969.

    Article  Google Scholar 

  26. P.E. Lush and T.M Cherry. The variational method in hydrodynamics. Quarterly Journal of Mechanics and Applied Mathematics, 9(1):6–21, 1956.

    Article  MATH  MathSciNet  Google Scholar 

  27. A.R. Manwell. A variational principle for steady homoenergic compressible flow with finite shocks. Wave Motion, 2(1):83–90, 1980.

    Article  MATH  MathSciNet  Google Scholar 

  28. P. Mauersberger. Erweitertes Hamiltonisches Prinzip in der Hydrodynamik. Gerlands Betraege Geophysics, 75(5):396–408, 1966.

    Google Scholar 

  29. P. Mauersberger. Variablentausch in Hamiltonischen Prinzip. Monatsberiehte Deutsches Akademie der Wissenschaften zu Berlin, 8(S.):289–298, 1966.

    MATH  Google Scholar 

  30. P. Mauersberger. Uber die Berucksichtigung freier Oberflachen in einer lokallen Fassung des Hamiltonischen Prinzip. Monatsberiehte Deutsches Akademie der Wissenschaften zu Berlin, 8(12):873–877, 1976.

    Google Scholar 

  31. H.K. Moffat. The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics, 35:117, 1969.

    Article  Google Scholar 

  32. I.G. Napolitano and C. Golia. Dual variational formulation of non-linear problem in fluid dynamics. In. 23 Congr. naz. assoc. ital. mecc. teor: ed appl. (Cagliari, 1976), sez. 43, Bologna, 1976.

    Google Scholar 

  33. P. Penfield. Hamilton’s principle for fluids. Journal of Physical Fluids, 9(6):1184–1194, 1966.

    Article  MATH  MathSciNet  Google Scholar 

  34. D. Rogula. Variational principle for material coordinates as dependent variables. application in relativistic continuum mechanics. Bulletin of the Academy of Polish Sciences; Services in Sciences and Technology, 8(10):781–785, 1970.

    Google Scholar 

  35. R.I. Seliger and G.B. Whitham. Variational principles in continuum mechanics. Proceedings of Royal Society, 305(ser. A):1–25, 1968.

    Article  MATH  Google Scholar 

  36. J. Serrin. Mathematical principles of classical fluid mechanics, volume VIII/I of Handbuch der physik. Berlin-Gottingen-Heidelberg, 1959.

    Google Scholar 

  37. M.J. Sewell. On reciprocal variational principle for perfect fluids. Journal of Mathematics and Mechanics, 12(4):495–504, 1963.

    MATH  MathSciNet  Google Scholar 

  38. D.A. Smith and C.V. Smith. When is Hamiltons principle an extremum principle. AIAA Journal, 12(11):1573–1576, 1974.

    Article  MATH  Google Scholar 

  39. J.J. Stephens. Alternate forms of the Herivel-Lin variational principle. Physics of Fluids, 10(1):76–77, 1967.

    Article  MATH  Google Scholar 

  40. B. Tabarrok and J.Z. Johnston. Some variational principles for isentropic and isothermal gas flows. Transaction on C.S.M.E., 3:187–192, 1975.

    Google Scholar 

  41. A. Taub. Hamilton’s principle for perfect compressible fluids. In Symposium in Applied Mathematics of American Mathematical Society, volume 1, pp. 148–157, 1949.

    MathSciNet  Google Scholar 

  42. K.I. Voljak. Variational principle for compressible fluid. Soviet Physics – Doklady, 22(10):562–563, 1977.

    MATH  Google Scholar 

  43. C.T. Wang. A note on Bateman’s variational principle for compressible fluid flow. Quarterly Applied Mathematics, 9:99–101, 1951.

    MATH  Google Scholar 

  44. C.T. Wang and G.V.R. Rao. A study of the non-linear characteristic of compressible flow equations by means of variational method. Journal of Aerosol Science, 17:343–348, 1950.

    MathSciNet  Google Scholar 

  45. V.E. Zakharov. Hamiltonian formalism for hydrodynamical models of plasma. Journal of Experimental and Theoretical Physics, 60(5):1714–1726, 1971.

    Google Scholar 

  46. V.E. Zakharov. The Hamiltonian formalism for waves in nonlinear media having dispersion. Radiophysics and Quantum Electronics, 17(4):326–343, 1974.

    Article  Google Scholar 

  47. G. Zemplen. Kriterien fur die physikalische Bedutung des unstetigen Losungen der hydrodynamischen Bewegungsgleichungen. Mathematical Annalen, 61:437–449, 1905.

    Article  MATH  MathSciNet  Google Scholar 

  48. V.A. Zhelnorovich. Toward the theory of constructing the continuum mechanics models. Soviet Physics – Doklady, 176(2), 1967.

    Google Scholar 

  49. V.A. Zhelnorovich and L.I. Sedov. On the variational method of derivation of the equations of state for a material medium and a gravitational field. Journal of Applied Mathematics and Mechanics (PMM), 42(5):827–838, 1978.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Wayne State University, 48202, Detroit, MI, USA

    V.L. Berdichevsky (Professor of Mechanics)

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  1. V.L. Berdichevsky
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Correspondence to V.L. Berdichevsky .

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Berdichevsky, V. (2009). Ideal Compressible Fluid. In: Variational Principles of Continuum Mechanics. Interaction of Mechanics and Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-88467-5_10

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  • DOI: https://doi.org/10.1007/978-3-540-88467-5_10

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  • Online ISBN: 978-3-540-88467-5

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