Nonclassical Dynamics of Classical Gases

  • M. S. Cramer
Part of the International Centre for Mechanical Sciences book series (CISM, volume 315)


In the present article we examine the dynamics of single-phase, equilibrium, i.e., classical, fluids in the dense gas regime. The behavior of fluids of moderately large molecular weight is seen to differ significantly from that of air and water under normal conditions. New phenomena include the formation and propagation of expansion shocks, sonic shocks, double sonic shocks, and shock-splitting. The more complicated existence conditions for shock waves are described and related to the dissipative structure. We also give a brief description of transonic flows and show that the critical Mach number for conventional blade shapes can be increased by a factor of 30–50% for these fluids.


Shock Wave Mach Number Compression Shock Shock Layer Entropy Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Thompson, P. A.: A fundamental derivative in gasdynamics, Phys. Fluids, 14 (1971), 1843–1849.CrossRefMATHGoogle Scholar
  2. 2.
    Borisov, A. A., AL. A. Borisov, S. S. Kutateladze and V. E. Nakorykov: Rarefaction shock wave near the critical-liquidvapour point, J. Fluid Mech., 126 (1983), 59–73.CrossRefGoogle Scholar
  3. 3.
    Novikov, I. I.: Existence of shock waves rarefaction, Dokl. Akad. Nauk SSSR, 59 (1948), 1545–1546.MATHGoogle Scholar
  4. 4.
    Kahl, G. D. and D. C. Mylin: Rarefaction shock possibility in a van der Waals-Maxwell fluid, Phys. Fluids, 12 (1969), 2283–2291.CrossRefGoogle Scholar
  5. 5.
    Temperley, H. N. V.: The theory of propagation in liquid helium II of ‘temperature waves’ of finite amplitude, Proc. Phys. Soc. Lond., A 64 (1951), 105–114.CrossRefMATHGoogle Scholar
  6. 6.
    Osborne, D. V.: Second sound in liquid helium II, Proc. Phys. Soc. Lond, A 64 (1951), 114–123.CrossRefGoogle Scholar
  7. 7.
    Dessler, A. J. and W. M. Fairbank: Amplitude dependence of the velocity of second sound, Phys. Rev., 104 (1956), 6–10.CrossRefGoogle Scholar
  8. 8.
    Turner, T. N. Second-sound shock waves and critical velocities in liquid helium II, Ph.D. Dissertation, California Institute of Technology, Pasadena, CA 1979.Google Scholar
  9. 9.
    Turner, T. N.: New experimental results obtained with second-sound shock waves, Physica, 107B (1981), 701–702.Google Scholar
  10. 10.
    Atkin, R. J., and N. Fox: The dependence of thermal shock wave velocity on heat flux in Helium II, J. Phys. C: Solid State Phys., 17 (1984), 1191–1198.CrossRefGoogle Scholar
  11. 11.
    Torczynski, J. R. Nonlinear fourth sound, Wave Motion, 7 (1985), 487–501.CrossRefMATHGoogle Scholar
  12. 12.
    Garrett, S. Nonlinear distortion of 4th sound in superfluid 3He-B, J. Acoust. Am., 60 (1981), 139–144.CrossRefGoogle Scholar
  13. 13.
    Bezzerides, B., D. W. Forslund, and E. L. Lindman: Existence of rarefaction shocks in a laser-plasma corona, Phys. Fluids, 21 (1978), 2179–2185.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Nariboli, G. A. and W. C. Lin: A new type of Burgers’ equation, ZAMM, 53 (1973), 505–510.CrossRefMATHGoogle Scholar
  15. 15.
    Kynch, G. J.: A theory of sedimentation, Trans. Faraday Soc., 48 (1952), 166–176.CrossRefGoogle Scholar
  16. 16.
    Shannon, P. T. and E. M. Tory: Settling of slurries, Ind. Engng Chem, 57 (1965), 18–25.Google Scholar
  17. 17.
    Amberg, G., A. A. Dahlkild, F. H. Bark, and D. S. Henningson: On time-dependent settling of a dilute suspension in a rotating conical channel, J. Fluid Mech., 166 (1986), 473–502.CrossRefMATHGoogle Scholar
  18. 18.
    Auzerais, F. M., R. Jackson,and W. B. R.ssel: The resolution of shocks and the effects of compressible sediments in transient settling, J. Fluid Mech., 195 (1988), 437–462.Google Scholar
  19. 19.
    Thompson, P. A., G. C. Carafano, and Y. G. Kim: Shock waves and phase changes in a large-heat-capacity fluid emerging from a tube, J. Fluid Mech., 166 (1986), 57–92.CrossRefMATHGoogle Scholar
  20. 20.
    Dettleff, G., G. E. A. Meier, H. D. Speckmann, P. A. Thompson, and C. Yoon: Experiments in shock liquefaction, In Proc. 13th Intl. Symp. on Shock Tubes and Waves (ed. C. E. Trainor zhaohuan J. G. Hall ), (1982), 716–723.Google Scholar
  21. 21.
    Thompson, P. A. and Y. G. Kim: Direct observation of shock splitting in a vapor-liquid system, Phys. Fluids, 26 (1986), 3211–3215.CrossRefGoogle Scholar
  22. 22.
    Thompson, P. A., H. Chaves, G. E. A. Meier, Y. G. Kim, and H. D. Speckmann: Wave splitting in a fluid of large heat capacity, J. Fluid Mech, 185 (1987), 385–414.CrossRefGoogle Scholar
  23. 23.
    Zel’dovich, Ya. B. and Yu. P. Raizer: Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Vol. 2, Academic 1967.Google Scholar
  24. 24.
    McQueen, R. G. and S. P. Marsh: Hugoniots of graphites of various initial densitites and the equation of state of carbon, in: Behavior of Dense Media Under High Dynamic Pressures, Gordon and Breach 1968, 207–216.Google Scholar
  25. 25.
    Gust, W. H. and D. A. Young: High Pressure Science and Technology, Vol. I, Plenum 1979.Google Scholar
  26. 26.
    Ivanov, A. G. and S. A. Novikov: Shock rarefaction waves in iron and steel, Zh. Eksp. Teor. Fiz, 40 (1961), 1880–1882.Google Scholar
  27. 27.
    Erkman, J. 0. Smooth spalls and the polymorphism in iron, J. Appl. Phys, 32 (1961), 939–944.CrossRefGoogle Scholar
  28. 28.
    Kolsky, H. Production of tensile shock waves in stretched natural rubber, Nature, 224 (1969), 1301.Google Scholar
  29. 29.
    Barker, L. M. and R. E. Hollenbach: Shock wave studies of PMMA, fused silica, and sapphire, J. Appl. Phys, 42 (1970), 42084226.Google Scholar
  30. 30.
    Bains, J. A. and M. A. Breazeale: Nonlinear distortion of ultrasonic waves in solids: approach of a stable backward sawtooth, J. Acoustic Soc. Amer., 57 (1975), 745–746.CrossRefGoogle Scholar
  31. 31.
    Conner, M. P. Shear Wave Measurements to Determine Nonlinear Elastic Response of Fused Silica Under Shock Loading, MS Thesis, Washington State University, Pullman, Washington.Google Scholar
  32. 32.
    Lee-Bapty, I. P. Nonlinear wave propagation in stratified and viscoelastic media, Ph.D. dissertation, Leeds University, England, 1981.Google Scholar
  33. 33.
    Morris, F. E. and G. A. Nariboli: Photoelastic waves, Int. J. Engng. Sci., 10 (1972), 765–774.CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Kakutani, T. and N. Yamasaki: Solitary waves on a two-layer fluid, J. Phys. Soc. Japan, 45 (1978), 674–679.CrossRefGoogle Scholar
  35. 35.
    Helfrich, K. R., W. K. Melville, and J. W. Miles: On interfacial waves over slowly varying topography, J. Fluid Mech., 149 (1984), 305–317.CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Bethe, H. A. The theory of shock waves for an arbitrary equation of state, Office Sci. Res. zhaohuan Dev. Rep. 545, Washington, D.C. 1942.Google Scholar
  37. 37.
    Hayes, W. D.: Gasdynamic discontinuities, in: Princeton Series on High Speed Aerodyanmics and Jet Propulsion, Princeton University Press 1960.Google Scholar
  38. 38.
    Beyer, R. T.: Nonlinear Acoustics, Naval Ship Systems Command, Dept. of the Navy, Washington, D. C. 1974.Google Scholar
  39. 39.
    Thompson, P. A.: Compressible Fluid Dynamics, McGraw-Hill, New York 1972.MATHGoogle Scholar
  40. 40.
    Zel’dovich, Ya. B.: On the possibility of rarefaction shock waves, Zh. Eksp. Teor. Fiz., 4 (1946), 363–364.Google Scholar
  41. 41.
    Thompson, P. A. and K. Lambrakis: Negative shock waves, J. Fluid Mech., 60 (1973), 187–208.CrossRefMATHGoogle Scholar
  42. 42.
    Lambrakis, K. and P. A. Thompson: Existence of real fluids with a negative fundamental derivative r, Phys. Fluids, 5 (1972), 933–935.CrossRefGoogle Scholar
  43. 43.
    Cramer, M. S.: Negative nonlinearity in selected fluorocarbons, Phys. Fluids, A. 1 (1989), 1894–1897.CrossRefGoogle Scholar
  44. 44.
    Martin, J. J. and Y. C. Hou: Development of an equation of state for gases, AIChE J., 1 (1955), 142–151.CrossRefGoogle Scholar
  45. 45.
    Rihani, D. N. and L. K. Doraiswany: Estimation of heat capacity of organic compounds from group contributions, Ind. Engr. Chem. Fund., 4 (1965), 17–21.CrossRefGoogle Scholar
  46. 46.
    Burnside, B. M.: Thermodynamic properties of five halogenated hydrocarbon vapour power cycle working fluids, J. Mechanical Engineering Science, 15 (1973), 132–143.CrossRefGoogle Scholar
  47. 47.
    Richter, H. R. D. and B. M. Burnside: A general programme for producing pressure-enthalpy diagram, J. Mechanical Engineering Science, 17 (1975), 31–39.CrossRefGoogle Scholar
  48. 48.
    Cramer, M. S. and R. Sen: Shock formation in fluids having embedded regions of negative nonlinearity, Phys. Fluids, 29 (1986), 2181–2191.CrossRefMATHGoogle Scholar
  49. 49.
    Cramer, M. S. and A. Kluwick: On the propagation of waves exhibiting both positive and negative nonlinearity, J. Fluid Mech., 142 (1984), 9–37.CrossRefMATHMathSciNetGoogle Scholar
  50. 50.
    Cramer, M. S. and R. Sen: Exact solutions for sonic shocks in van der Waals gases, Phys. Fluids, 30 (1987), 377–385.CrossRefMATHGoogle Scholar
  51. 51.
    Gilbarg, D.: The existence and limit behavior of the one-dimensional shock layer, Am. J. Maths, 73 (1951), 256–274.CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Cramer, M. S.: Shock splitting in single-phase gases, J. Fluid Mech., 199 (1989), 281–296.CrossRefMATHMathSciNetGoogle Scholar
  53. 53.
    Menikoff, R. and B. Plohr: Riemann problem for fluid flow of real materials, Reviews of Modern Physics, 61 (1989), 75–130.CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    Lax, P. D.: Shock waves and entropy, in: Contributions to Nonlinear Functional Analysis (ed. E. H. Zarantonello ), Academic 1971.Google Scholar
  55. 55.
    Cramer, M. S., A. Kluwick, L. T. Watson, and W. Pelz: Dissipative waves in fluids having both positive and negative nonlinearity, J. Fluid Mech., 169 (1986), 323–336.CrossRefMATHGoogle Scholar
  56. 56.
    Cramer, M. S.: Structure of weak shocks in fluids having embedded regions of negative nonlinearity, Phys. Fluids, 30 (1987), 3034–3044.CrossRefGoogle Scholar
  57. 57.
    Taylor, G. I.: The conditions necessary for discontinuous motion in gases, Proc. R. Soc. Lond., A 84 (1910), 371–377.CrossRefMATHGoogle Scholar
  58. 58.
    Lee-Bapty, I. P. and D. G. Crighton: Nonlinear wave motion governed by the modified Burger’s equation, Phil. Trans. R. Soc. Lond., A 323 (1987), 173–209.CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    Reid, R. C. J. M. Prausnitz, and B. E. Poling: The Properties of Gases and Liquids, 4th Edition, Wiley 1987.Google Scholar
  60. 60.
    Landau, L. D. and E. M. Lifshitz: Fluid Mechanics, Addison-Wesley 1959.Google Scholar
  61. 61.
    Cramer, M. S. Nonclassical Dynamics of Classical Gases, Virginia Polytechnic Institute and State University Engineering Report #VPI-E-89–20, Blacksburg, VA 1989.Google Scholar

Copyright information

© Springer-Verlag Wien 1991

Authors and Affiliations

  • M. S. Cramer
    • 1
  1. 1.Virginia Polytechnic Inst. and State UniversityBlacksburgUSA

Personalised recommendations