Fluid Dynamics

, Volume 52, Issue 2, pp 321–328 | Cite as

Propagation of discontinuities against a static background

  • A. N. Golubyatnikov
  • S. D. Kovalevskaya


The solution of the ideal gasdynamic equations describing propagation of a shock wave initiated, for example, by the motion of a piston against an inhomogeneous static background is considered. The solution is constructed in the form of Taylor series in a special time variable which is equal to zero on the shock wave. In the case of weak shock waves divergence of the series serves as the constraint for such an approach. Then the solution is constructed by linearizing the equations about the solution with a weak discontinuity. In the case of a given background the last solution can be always found exactly by solving successively a set of transport equations, all these equations are reduced to linear ordinary differential equations. The presentation begins from the one-dimensional solutions with plane waves and ends by discussion of spatial problems.


shock wave Taylor series weak discontinuity transport equation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    L. I. Sedov, Similarity and Dimensional Methods in Mechanics (Nauka, Moscow, 1981; CRC Press, Boca Raton, 1993).Google Scholar
  2. 2.
    L. D. Landau, “Shock Waves at Far Distances from the Place of their Initiation,” Prikl. Mat. Mekh. 9, No. 4, 286–292 (1945).Google Scholar
  3. 3.
    L. Crussard, “Sur la Deformation des Ondes dans les Gas et sur les Iterferences Finies.” C. r. Acad. Sci. 156, No. 6, 447–450 (1913); Sur la Propagation et l’Alteration des Ondes de Choc,” Ibid., No. 8, 611–613 (1913).MATHGoogle Scholar
  4. 4.
    V. P. Korobeinikov, “Damping ofWeak Magnetohydrodynamic Shock Waves,” Magnitnaya Gidrodinamika, No. 2, 25–30 (1967).Google Scholar
  5. 5.
    L. I. Sedov, Continuum Mechanics, Vol. 1 (Nauka, Moscow, 1970) [in Russian].MATHGoogle Scholar
  6. 6.
    A. N. Golubyatnikov and S. D. Kovalevskaya, “On the Acceleration of Shock Waves in an Electric Field,” in: Proceedings of XIth International Scientific Conference “Modern Problems of Electrophysics and Electrodynamics,” June 29–July 3, 2015, Saint-Petersburg (Izd. Dom “Petrogradskii,” Saint-Petersburg, 2015) [in Russian], pp. 78–80.Google Scholar
  7. 7.
    A. G. Kulikovskii and G. A. Lyubimov, Magneto-Hydrodynamics (2nd Edition, Logos,Moscow, 2005; 1st Edition, Addison-Wesley, Reading, Mass., 1965).Google Scholar
  8. 8.
    G. G. Chernyi, Gas Dynamics (Nauka, Moscow, 1988) [in Russian].Google Scholar
  9. 9.
    B. L. Rozhdestvenskii and N. N. Yanenko, Systems of Quasilinear Equations (Nauka, Moscow, 1978) [in Russian].MATHGoogle Scholar
  10. 10.
    A. N. Golubyatnikov and S. D. Kovalevskaya, “ShockWave Acceleration in a Magnetic Field,” Fluid Dynamics 49 (6), 844–848 (2014).MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. N. Golubyatnikov and S. D. Kovalevskaya, “Acceleration of Weak Shock Waves,” Fluid Dynamics 50 (5), 705–710 (2015).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    P. Yu. Georgievskii, V. A. Levin, O. G. Sutyrin, “Interaction between a ShockWave and a Longitudinal Low-Density Gas Layer,” Fluid Dynamics 51 (5), 696–702 (2016).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    T. A. Zhuravskaya and V. A. Levin, “Stability of Gas Mixture Flow with a Stabilized Detonation Wave in a Plane Channel with a Constriction,” Fluid Dynamics 51 (4), 544–551 (2016).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Faculty of Mechanics and MathematicsLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations