Nonlinear Wave Propagation Through a Radiating van der Waals Fluid with Variable Density

  • Madhumita GangopadhyayEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


We examine a quasilinear system of PDEs governing the one-dimensional unsteady flow of a radiating van der Waals fluid in radial, cylindrical and spherical geometry. The local value of the fundamental derivative (Γ) associated with the medium is of order O(𝜖) and changes sign about the reference state (Γ = 0); the undisturbed medium is assumed to be spatially variable. An asymptotic method is employed to obtain a transport equation for the system of Navier Stokes equations; the impact of radiation and the van der Waals parameters on the evolution of the initial pulse is studied.


Hyperbolic system Mixed nonlinearity van der Waals fluid Radiation 



The author wishes to sincerely thank the University Grants Commission, India, for its support through a Major Research Project No. F/788/2012/SR.


  1. 1.
    Bethe, H.A.: On the theory of shock waves for an arbitrary equation of state. Tehnical Report No. 545, Office of Scientific Research and Development (1942)Google Scholar
  2. 2.
    Clarke, J.F., McChesney, M.: Dynamicss of Relaxing gases. Butterworth, London (1976)Google Scholar
  3. 3.
    Fan, H., Slemrod, M.: Dynamic flows with liquid/vapor phase transitions. In: Handbook of Mathematical Fluid Dynamics, 1, pp. 373–420, North Holland, Amsterdam (2002)CrossRefGoogle Scholar
  4. 4.
    Fusco, D.: Some comments on wave motions described by nonhomogeneous quasilinear first order hyperbolic systems. Meccanica 17, 128–137 (1982)CrossRefGoogle Scholar
  5. 5.
    Kluwick, A., Cox, E.A.: Nonlinear waves in materials with mixed nonlinearity. Wave Motion 27, 23–41 (1998)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Pai, S.I.: Radiation Gasdynamics. Springer, New York (1966)CrossRefGoogle Scholar
  7. 7.
    Penner, S.S., Olfe, D.B.: Radiation and Reentry. Academic Press, New York (1968)Google Scholar
  8. 8.
    Radha, Ch., Sharma, V.D.: High and low frequency small amplitude disturbances in a perfectly conducting and radiating gas. Int.J.Engng Sci. 33, 2001–2010 (1995)CrossRefGoogle Scholar
  9. 9.
    Sharma, V.D., Madhumita G.: Nonlinear wave propagation through a stratified atmosphere. J. Math. Anal. Appl. 311, 13–22 (2005)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Shukla, T.P., Sharma, V.D.: Weakly nonlinear waves in nonlinear fluids. Studies in Applied Mathematics 00, 1–22 (2016)Google Scholar
  11. 11.
    Shukla, T.P., Madhumita G., Sharma, V.D.: Evolution of planar and cylindrically symmetric magneto-acoustic waves in a van der waals fluid. Int.J.Nonlinear Mechanics 91, 58–68 (2017)CrossRefGoogle Scholar
  12. 12.
    Thompson, P.A.: A fundamental derivative in gas dynamics. Phys. Fluids 14, 1843–1849 (1971)CrossRefGoogle Scholar
  13. 13.
    Thompson, P.A., Lambrakis, K.S.: Negative shock waves. J.Fluid Mech. 60, 187–207 (1973)CrossRefGoogle Scholar
  14. 14.
    Varley, E., Cumberbatch, E., Nonlinear high frequency sound waves. J. Inst. Math. Appl. 2, 133–143 (1966)CrossRefGoogle Scholar
  15. 15.
    Vincenti, W.G., Kruger, C.H.: Introduction to Physical Gasdynamics. Wiley. New York (1965)Google Scholar
  16. 16.
    Zhao, N., Mentrelli, A., Ruggeri, T., Sugiyama, M.: Admissible shock waves and shock induced phase transitions in a van der waals fluid. Phys. Fluids 23, 086101 (2011)CrossRefGoogle Scholar
  17. 17.
    Zeldovich, Y.B.: On the possibility of rarefaction shock waves. Zh. Eksp. Teor. Fiz. 4, 363–364 (1946)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MumbaiMumbaiIndia

Personalised recommendations