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Some Mathematical Aspects of Rarefied Gasdynamics as Applied to Hypersonics, Reentry and Magneto-Gas-Dynamics

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Dinamica dei gas rarefatti

Part of the book series: C.I.M.E. Summer Schools ((CIME,volume 33))

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Abstract

There exists no precise definition of the boundaries of the regime known under the name of hypersonics. On one side it includes the. reentry phenomena which are inseparably connected with all the domains of the classical aero-and fluid-dynamics, involving all of the classical regions of the sub-, trans-, and supersonic nature. On the other side it involves the relativistic phenomena with the velocity of light being another barrier, the light-barrier, having a some sort of analogy to the old sonic barrier. Practically, the hypersonics is interested in the sub-light regime, leaving the super-light region still in the sphere of speculations of the theoretical physics. Hypersonics applies all the possible tools, ever invented, developed and worked out by the mechanics of continuous media, kinetic theory of gases, Newtonian free molecule technique and finally the classical special theory of relativity. This refers to all kinds of gaseous media, i.e., ideal, perfect, real, ionized, etc. There is no limit in that respect from any point of view. Concerning the problem of solving the differential and integral equations occurring in the fields in question, there are used all of the possible techniques and methods known in the theory of partial and ordinary differential as well as of integro-differential equations. One can mention here the classical techniques, special functions, algebraic, integral operator, topological techniques, reduction of the number of independent variables, etc. Each of these methods possesses certain advantages and disadvantages.

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List of References

  1. Aziz, A. K., Gilbert, R. P., Howard, H. C. : On a Non-linear Elliptic Boundary Value Problem with Generalized Goursat Data. Tech. Note BN-377, Univ. of Maryland, Sept. 1964.

    Google Scholar 

  2. Bergman, S. : Two-Dimensional Transonic Flow Patterns. Amer.J.Math.70, 1948, pp. 856–891.

    Article  MATH  MathSciNet  Google Scholar 

  3. Bergman, S. : On Solutions of Linear Partial Differential Equations of Mixed Type. Amer. J. Math. 74, 1952, pp.444–474.

    Article  MATH  MathSciNet  Google Scholar 

  4. Bergman, S. : Integral Operators in the Theory of Linear Partial Differential Equations. Springer-Verlag, Berlin-Goettingen-Heidelberg, 1961.

    MATH  Google Scholar 

  5. Bergman, S. : On Value Distribution of Meromorphic Functions of Two Complex Variables. Studies in Mathematical Analysis and Related Topics. (Ed. by Gilbarg, D., Solomon, H, et al.), Stanford Univ. Press, 1962.

    Google Scholar 

  6. Bergman, S. : On Distortion Theorems in the Theory of Quasi-Pseudo Conformal Mappings. Applied Mathematics and Statistics Laboratories, Stanford University, Feb., 1963.

    Google Scholar 

  7. Bergman, S. : On Integral Operators Generating Stream Functions of Compressible Fluids, Non-Linear Problems in Engineering. Academic Press, Inc., 1964 (to appear).

    Google Scholar 

  8. Bergman, S., Bojanić, R. : Application of Integral Operators to the Theory of Partial Differential Equations with Singular Coefficients. Archive for Rational Mechanics and Analysis, Vol. 10, No. 4, 1962, pp. 323–340.

    Article  MATH  MathSciNet  Google Scholar 

  9. Bergman, S., Schiffer, M. : Kernel Functions and Elliptic Differential Equations in Mathematical Physics. Academic Press, Inc., New York, 1953.

    MATH  Google Scholar 

  10. Blank, A. A., Friedrichs, K. O., Grad, H. : Notes on Magneto-hydrodynamics, V. Theory of Maxwell's Equations without Displacement Current. Physics and Mathematics, NYO-6486, AEC Computing and Applied Mathematics Center ; Institute of Mathematical Sciences, New York University, Nov. 1, 1957.

    Google Scholar 

  11. Burnett, D. : The Distribution of Molecular Velocities and the Mean Motion in a Non-uniform Gas. Proc. London Math. Soc., Vol.40, 1935, pp.382.

    Article  Google Scholar 

  12. Chapman, S., Cowling, T. G. : The Mathematical Theory of Non-uni-form Gases, 1st ed. Cambridge University Press, London, 1939.

    Google Scholar 

  13. Courant, R., Friedrichs, K. O. : Supersonic Flow and Shock Waves. Interscience Publishers, Inc., New York, 1948.

    MATH  Google Scholar 

  14. Eichler, M. M. E. : On The Differential Equation uxx +uyy + + N(x)u = 0. Trans. Amer. Math. Soc, Vol. 65,1949, pp.259–278.

    MATH  MathSciNet  Google Scholar 

  15. Epstein, P. S. : On the Resistance Experienced by Spheres in their Motion Through Gases. Phys. Review, Vol.23, 1924, pp.710.

    Article  Google Scholar 

  16. Ergelyi, A. et al. : Higher Transcental Functions, Vol.1, II. McGraw-Hill Pub. Co., New York, 1953.

    Google Scholar 

  17. Gilbert, R. P. : On the Singularities of Generalized Axially Symmetric Potentials. Archive for Rational Mechanics and Analysis, Vol.6, No. 2, 1960, pp. 171–176.

    Article  MATH  MathSciNet  Google Scholar 

  18. Gilbert, R. P. : Singularities of Three-Dimensional Harmonic Functions. Pacific J. Math., Vol.10, No.4, 1960,pp. 1243–1255.

    MATH  MathSciNet  Google Scholar 

  19. Gilbert, R. P. : On the Geometric Character of Singularity Manifolds for Harmonic Functions in Three Variables, I. Tech. Note BN-256,AFOSR-1045, Univ. of Maryland, Sept., 1961. Also Archive for Rational Mechanics and Analysis, Vol.9, No.4, 1962, pp. 352–360.

    Google Scholar 

  20. Gilbert, R. P. : A note on Harmonic Functions in (p+2) Variables. Archive for Rational Mechanics and Analysis Vol. 8, No. 3, 1961, pp.223–227.

    Article  MATH  MathSciNet  Google Scholar 

  21. Gilbert, R. P. : On Harmonic Functions of Four Variables with Rational p4 - Associates. Tech. Note BN-274, AFOSR-2252, Univ. of Maryland, Jan., 1962.

    Google Scholar 

  22. Gilbert, R. P. : Poisson's Equation and Generalized Axially Symmetric Potential Theory. Tech. Note BN-283, AFOSR-2450, Univ. of Maryland, March, 1962.

    Google Scholar 

  23. Gilbert, R. P. : A Note on the Singularities of Harmonic Functions in Three Variables. Proc. Amer. Math. Soc, Vol.13, No. 2, April 1962, pp.229–232.

    Article  MATH  MathSciNet  Google Scholar 

  24. Gilbert, R. P. : Some Properties of Generalized Axially Symmetric Potentials. Amer.J. Math., Vol.LXXXIV, No. 3, July, 1962, pp. 475–484.

    Article  Google Scholar 

  25. Gilbert, R. P. : Harmonic Functions in Four Variables with Algebraic and Rational p4 Associates. Tech. Note BN-294, Univ. of Maryland. July, 1962.

    Google Scholar 

  26. Gilbert, R. P. : Composition Formulas in Generalized Axially Symmetric Potential Theory. Tech. Note BN-298, Univ. of Maryland, Oct., 1962.

    Google Scholar 

  27. Gilbert, R. P. : Operators Which Generate Harmonic Functions in Three Variables. Tech. Note BN-306, Univ. of Maryland, Dec, 1962.

    Google Scholar 

  28. Gilbert, R. P. : Bergman's Integral Operator Method in Generalized Axially Symmetric Potential Theory (u) NOLTR 63–124, United States Naval Ordnance Laboratory, White Oak, Maryland, June 14, 1963,

    Google Scholar 

  29. Gilbert, R. P. : On Solutions of the Generalized Axially Symmetric Wave Equation Represented by Bergman Operators. Tech. Note BN-350, Univ. of Maryland, March, 1964.

    Google Scholar 

  30. Gilbert, R. P. : On Generalized Axially Symmetric Potentials Whose Associates are Distributions. Tech. Note BN-356, Univ. of Maryland, April, 1964.

    Google Scholar 

  31. Gilbert, R. P. : Composition Formulas in Generalized Axially Symmetric Potential Theory. J. Math. and Mech., Vol.13, No. 4, 1964, pp.557–588.

    Google Scholar 

  32. Gilbert, R. P., Howard, H. C. : On Solutions of the Generalized Bi-Axially Symmetric Helmholtz Equation Generated by Integral Operators. Tech. Note BN-352, Univ. of Maryland, April, 1964.

    Google Scholar 

  33. Gilbert, R. P., Howard, H. C. : Integral Operator Methods for Generalized Axially Symmetric Potentials in (n+1) Variables. Tech. note BN-366, Univ.of Maryland, July, 1964.

    Google Scholar 

  34. Grad H. : On the Kinetic Theory of Rarefied Gases. Comm.Pure Appl. Math., Vol.2, 1949, pp.331–407.

    Article  MATH  MathSciNet  Google Scholar 

  35. Grad, H. : Asymptotic Theory of the Boltzmann Equation, I.Phys. Fluids, Vol.6, pp.147.

    Google Scholar 

  36. Grad, H. : Asymptotic Theory of the Boltzmann Equation, II. Rarefied Gas Dynamics. Proc. of the Third International Symposium on Rarefied Gas Dynamics, Held at the Palais de 1'Unesco, Paris, in 1962, Vol.1; Supplement 2, Academic Press, 1963. pp.26–59.

    MathSciNet  Google Scholar 

  37. Heinema, M. : Theory of Drag in Highly Rarefied Gases. Comm. Pure Appl. Math., Vol.1, no. 3, 1948, pp.259–273.

    Article  Google Scholar 

  38. Henrici, P. : Zur Funktionentheorie del Wellengleichung Commentarii Mathematici Helvetici, Vol. XXVII, 1953, pp. 235–293.

    Article  MathSciNet  Google Scholar 

  39. Henrici, P. : Complete Systems of Solutions for a Class of Singular Elliptic Partial Differential Equations. Boundary Problems in Differential Equations. Univ. of Wis. Press, Madison, Wis., 1960,pp. 19–34.

    Google Scholar 

  40. Ikenberry, E., Truesdell, C. : On the Pressures and the Flux of Energy in a Gas according to Maxwell's Kinetic Theory, I, II, J. Rational Mechanics and Analysis, Vol.5, No. 1, Jan., 1956, pp.1–54, 55–128.

    MATH  MathSciNet  Google Scholar 

  41. Jaffè, G. : Zur Methodik der Kinetischen Gastheorie. Annalen der Physik, Ser.5, Vol.6, 1930, pp.195–252.

    Article  Google Scholar 

  42. Keller, J. B. : On the Solution of The Boltzmann Equation for Rarefied Gases. Comm. Pure Appl. Math., Vol.1, No. 3, 1948, pp 275–285.

    Article  Google Scholar 

  43. Kreyszig, E. : Coefficient Problems in Systems of Partial Differential Equations. Arch.Rat. Mech. Anal., Vol.1, 1958, pp. 283–294.

    Article  MATH  MathSciNet  Google Scholar 

  44. Kreyszig, E. : On Singularities of Solutions of Partial Differential Equations in Three Variables. Arch. Rat. Mech. Anal., Vol.2, 1958, pp. 151–159.

    Article  MATH  MathSciNet  Google Scholar 

  45. Kreyszig, E. : On Regular and Singular Harmonic Functions of Three Variables. Arch. Rat. Mech. Anal., Vol. 4, 1960, pp. 352– 370.

    Article  MATH  MathSciNet  Google Scholar 

  46. Kreyszig, E. : Kanonische Integral Operatoren zur Er-zeugung Harmonischer Funktionen von vier Veraenderlichen. Arch, der Math., Vol.14, 1963, pp. 193–203.

    Article  MATH  MathSciNet  Google Scholar 

  47. Krzywoblocki, v. M. Z. : Bergman's Linear Integral Operator Method in the Theory of Compressible Fluid Flow. Oesterreich. Ing.-Arch., Vol. 6, 1952, pp. 330–360; Vol. 7,1953, pp. 336–370; Vol.8, 1954, pp. 237–263; Vol.10, 1956, pp. 1–38.

    MATH  MathSciNet  Google Scholar 

  48. Krzywoblocki, v. M. Z. : On the Generalized Integral Operator Method in the Subsonic Diabatic Flow of a Compressible Fluid. Proc. of the IX International Congress of Applied Mechanics, Brussells, 1956, Paper I–II, 1957, pp. 414–419.

    Google Scholar 

  49. Krzywoblocki, v. M. Z. : On the Generalized Theory of the Laminar Two-Dimensional Boundary Layer Along a Flat Plate in Continuum and Slip Flow Regimes, I, II. Bulletin de la la Société Mathémat. de Grèce 1, Vol. 29, 1,2, 3, 1954, pp. 34–74; III. Vol. 31, 1,2,3, 1959, pp. 41–68.

    Google Scholar 

  50. Krzywoblocki, v. M. Z. : On a Method of Solving the General System of Equations in Magneto-Gas-Dynamics. Bull, de la Soc. Mathématique de Grèce, Nouvelle Serie, Tome 1, Fasc. 1, 1960, pp. 63–97.

    MATH  Google Scholar 

  51. Krzywoblocki, v. M. Z. : Bergman's Linear Integral Operator Method in the Theory of Compressible Fluid Flow. (With an appendix by Davis, P., Rabinovitz, P.). Springer-Verlag, Vienna, 1960.

    MATH  Google Scholar 

  52. Krzywoblocki, v. M. Z. : On the Fundamentals of the Theories of Relativity. Research Report, R-61–33, The Martin Co., Denver, U. S. A., October 1961, pp.ii + 26.

    Google Scholar 

  53. Krzywoblocki, v. M. Z. : On the General Form of the Special Theory of Relativity. Parts I to IV. Acta Physica Austriaca, Vol.8, No. 4, 1960, pp. 387–394; Vol.14, No. 1, 1961, pp. 22–28; Vol. 14, No. 1, 1961, pp. 39–49;Vol. 14, No. 2, 1962, pp. 239–241.

    Google Scholar 

  54. Krzywoblocki, v. M. Z. : Special Relativity- A Particular Energy Formulation in Newtonian Mechanics. Part I, II. Acta Physica Austriaca, Vol. 15, No. 3, pp. 201–212; Vol, 15, N. 3, 1962, pp. 251–261.

    MathSciNet  Google Scholar 

  55. Krzywoblocki, v. M. Z. : On the Fundamentals of the Relativistic Theories. Acta Physica Austriaca, Vol. 15, No.4, 1962, pp.320–336.

    MATH  MathSciNet  Google Scholar 

  56. Krzywoblocki, v. M. Z. : Ergodic Problem in the Theory of Vibrations and Wave Propagation. Zaga-dnienia Dragán Nieliniow., Vol.4, 1962, pp. 53–76.

    MATH  Google Scholar 

  57. Krzywoblocki, v. M. Z. : Generalization of Topological Aspects in Existence Proofs to Magnetohydrody-namics. Fundamental Topics in Relativistic Fluid Mechanics and Magnetohydro-dynamics. Ed. by Wasserman, R., Wells, C.P. Proc. of a Symposium Held at Michigan State Univ., Oct., 1962. Published by Academic Press, New York, 1963, pp. 91–124.

    Google Scholar 

  58. Krzywoblocki, v. M. Z. : Some Ergodic Problems in Physical Mathematics. Proc. Mountain State Navy Res. and Devel. Clinic, Raton, New Mexico, Sept. 28–29, 1961, Published by Commun. and Electronics Foundation, Raton, New Mexico, 1963, pp.F35–F66.

    Google Scholar 

  59. Krzywoblocki, v. M. Z. : Operators in Ordinary Differential Equations. J. fuer Reine Angew. Mathe-matik, Vol. 214/215, 1964, pp. 137–140.

    Article  Google Scholar 

  60. Krzywoblocki, v. M. Z. Hassan, H.A. : Bergman's Linear Integral Operator Method in Diabatic Flow. J. Soc. Indust. Appl. Math., Vol. 5, 1957, pp. 47–65.

    Article  MATH  MathSciNet  Google Scholar 

  61. Krzywoblocki, v. M. Z. Hassan, H. A. : On the Limiting Lines in Diabatic Flow. Seorsum Impressum Ex Tom. VI Com-mentariorum Mathematicorum Universitatis Sancti Pauli, Tokyo, 3,1, 1958, pp. 115–139.

    Google Scholar 

  62. Krzywoblocki, v. M. Z. Roth, H. : On the Reduction of the Number of In-dependent Variables in Systems of Partial Differential Equations. (To be published).

    Google Scholar 

  63. Loeb, L. B. : Kinetic Theory of Gases. McGraw-Hill Pub. Co., Inc., New York, 1934.

    Google Scholar 

  64. Maxwell, J. C. : On the Dynamical Theory of Gases. Phil. Trans. R.Soc. London, 157 (1866), 49–88. (With corrections) Phil. Mag. (4) 35, 129–145, 185–217(1868). Papers, 2, 26–78.

    Google Scholar 

  65. Michal, A. D. : Differential Invariants and Invariant Partial Differential Equations in Normed Linear Spaces. Proc. Natl. Acad. Sci., U.S. 37, No.9, 1952, pp. 623–627.

    Article  MathSciNet  Google Scholar 

  66. Mises, V. R., Schiffer, M. : Advances in Applied Mechanics, Vol.1, 1948.

    Google Scholar 

  67. Morgan, A. J. A. : The Reduction by One of the Number of Indepentent Variables in Some Systems of Partial Differential Equations. Quart. J. Math. [N.S] 3, No. 12, 1952, pp. 250–259.

    Article  MATH  Google Scholar 

  68. Ringleb, F. : Exakte Loesungen der Differentialgleichungen einer adiabatischen Gasstroemung. Z. Angew. Math. Mech., Vol. 20, 1940 pp. 185–195. Also, Aero. Sci J., 1942.

    Article  MathSciNet  Google Scholar 

  69. Sanger, E. : Gaskinetik sehr hoher Fluggeschwindi-gkeiten. Deutsche Luftfahrtforschung, Bericht 972, Berlin, 1938.

    Google Scholar 

  70. Schaaf, S. A., Chambre, P. L. : Flow of Rarefied Gases. High speed Aerodynamics and Jet Propulsion. Vol. III, Fundamentals of Gas Dynamics (Ed. Emmons, H.W.). Priceton Univ. Press, 1958.

    Google Scholar 

  71. Stark, J. M. : Transonic Flow Patterns Generated by Bergman's Integral Operator. Dept. of Math. Stanford University, Aug., 1964.

    Google Scholar 

  72. Tsien, H. S. : Superaerodynamics, Mechanics of Rarefied Gases, J. of the Aeronautical Sciences, Dec., 1946, pp. 653–664.

    Google Scholar 

  73. Zahm, A. F. : Superaerodynamics. J. of the Franklin Institute, Vol, 217, 1934, pp. 153–166.

    Article  MATH  Google Scholar 

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  1. Michigan State University, East Lansing, MI, USA

    M. Z. v. Krzywoblocki

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Krzywoblocki, M.Z. (2011). Some Mathematical Aspects of Rarefied Gasdynamics as Applied to Hypersonics, Reentry and Magneto-Gas-Dynamics. In: Ferrari, C. (eds) Dinamica dei gas rarefatti. C.I.M.E. Summer Schools, vol 33. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11024-5_2

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