An Exact Solution for the Propagation of Shock Waves in Self-Gravitating Perfect Gas in the Presence of Magnetic Field and Radiative Heat Flux

An Exact Solution for the Propagation of Shock Waves in Self-Gravitating Perfect Gas in the Presence of Magnetic Field and Radiative Heat Flux

G. Nath Mrityunjoy Dutta R. P. Pathak

Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Allahabad - 211004, India.

Department of Mathematics, National Institute of Technology, Raipur G. E. Raipur-492001, India.

Corresponding Author Email: 
gnath@mnnit.ac.in; gn_chaurasia_univgkp@yahoo.in
Page: 
907-927
|
DOI: 
https://doi.org/10.18280/mmc_b.860406
Received: 
5 October 2017
|
Accepted: 
20 November 2017
|
Published: 
31 December 2017
| Citation

OPEN ACCESS

Abstract: 

Propagation of spherical shock wave with azimuthal magnetic field and radiation heat flux in self-gravitating perfect gas is investigated. The azimuthal magnetic field and the initial density are assumed to vary according to power law. An exact similarity solution is reported when loss of energy due to radiation escape is notable and radiation pressure is non-zero. The entire energy of the shock wave is varying and increases with time. The effects of variation of the radiation pressure number, the initial density variation index, the Alfven-Mach number, the gravitational parameter and the adiabatic exponent are workout in detail. The shock strength increases with an increase in the initial density variation index. On the other hand, presence of magnetic field or an increment in the value of the radiation pressure number or the ratio of specific heats or gravitational parameter the shock strength decreases. It is obtained that increase in the radiation pressure number and gravitational parameter has same behavior on the flow variables. Also, it is observed that an increase in the value of gravitational parameter and the adiabatic exponent have same behavior on the fluid velocity, the material pressure, the radiation pressure, the mass and the radiation flux and azimuthal magnetic field.

Keywords: 

MHD Shock waves, Similarity solution, Self-gravitating perfect gas, Radiation pressure and radiation energy, Radiation heat flux.

1. Introduction
2. Fundamental Equations of Motions and Boundary Conditions
3. Self-similarity Transformations
4. Results and Discussion
Conclusion
Nomenclature
  References

1. L.I. Sedov, Similarity and dimensional methods in mechanics, 1959, Academic Press, New York, NY, USA.

2. G. Taylor, The air wave surrounding an expanding sphere, 1946, Proc. R. Soc. Lond. A, vol. 186, pp. 273-292.

3. G. Taylor, The formation of a blast wave by a very intense explosion I, theoretical discussion, 1950, Proc. R. Soc. Lond. A, vol. 201, pp. 159-174.

4. P. Carrus, P. Fox, F. Hass, Z. Kopal, The propagation of shock waves in a steller model with continuous density distribution, 1951, Astrophys. J., vol. 113, pp. 496–518.

5. S.C. Purohit, Self-similar homothermal flow of self gravitating gas behind shock wave, 1974, J. Phys.  Soc. (Japan), vol. 36, pp. 288–292.

6. O. Nath, S. Ojha, H.S. Takhar, A study of stellar point explosion in a self-gravitating radiative magnetohydrodynamic medium, 1991, Astrophys. Space Sci., vol. 183, pp. 135–145.

7. G. Nath, A.K. Sinha, A self-similar flow behind a magnetogasdynamics shock wave generated by a moving piston in a gravitating gas with variable density: isothermal flow. Phys. Res. Inter 2011:

8.  A. Sakurai, Propagation of spherical shock waves in stars, 1956, J. Fluid Mech.1, pp. 436–453.

9. M.H. Rogers, Analytic solutions for blast wave problem with an atmosphere of varying density, 1957, Astrophys. J., vol. 125, pp. 478–493.

10. P. Rosenau, S. Frankenthal, Equatorial propagation of axisymmetric magnetohydrodynamic shocks, 1976, I. Phys. Fluids, vol. 19, pp. 1889–1899.

11. J.P. Vishwakarma, A.K. Yadav, Self-similar analytical solutions for blast waves in inhomogeneous atmospheres with frozen-in-magnetic field, 2003, Eur. Phys. J.B., vol. 34, pp. 247–253.

12. G. Nath, Magnetogasdynamic shock wave generated by a moving piston in a rotational  axisymmetric isothermal flow of perfect gas with variable density, 2011, Adv. Space Res., vol. 47, pp. 1463–1471.

13. G. Nath, Unsteady isothermal flow behind a magnetogasdynamic shock wave in a self-gravitating gas with exponentially varying density, 2014, J. Theor. Appl. Phys., vol. 8, pp. 1-8.

14. G.Nath and J.P.Vishwakarma, Propagation of a strong spherical shock wave in a gravitating or non-gravitating dusty gas with exponentially varying density, 2016 Acta Astronautica., vol. 123, pp. 200-2013.

15. K.C Wang, The piston problem with thermal radiation, 1964, J. Fluid Mech., vol. 20, pp. 447–455.

16. R. E. Marshak, Effects of radiation on shock wave behavior. Phys. Fluids, 1, 24-29. 1958.

17. L.A. Elliot, Similarity methods in radiation hydrodynamics, 1960, Proc. Roy. Soc. A., vol. 258, pp. 287-301.

18. S. Ashraf, P.L. Sachdev, An exact similarity solution in radiation-gas-dynamics, 1970, Proc. Indian Acad. Sci. A, vol. 71, pp. 275-281.

19. B.G. Verma and V.P.Vishwakarma, An exact similarity solution for spherical shock wave in magnetoradiative gas, 1978, Astrophys. Space Sci., vol. 58, pp. 139-147.

20. J.P. Vishwakarma, A.K. Maurya, A.K. Singh, Cylindrical shock waves in a non-ideal gas with radiation heat-flux and magnetic field, 2011, AMSE Journals, Modelling B, vol. 80, pp. 35-52.

21. J.S. Shang, Recent research in magneto-aerodynamics, 2001, Prog. Aerosp. Sci., vol. 21, pp. 1-20.

22. R.M. Lock, A.J. Mestel, Annular self-similar solution in ideal gas magnetogasdynamics, 2008, J. Fluid Mech., vol. 74, pp. 531-554.

23. J.P. Vishwakarma, R.C. Shrivastava, A. Kumar, An Exact similarity solution in radiation Magneto gas dynamics for the flows behind a spherical shock, 1987, Astrophys. Space Sci. vol. 129, pp. 45-52.

24. G.C. Mc. Vittie, Spherically solutions of the equations of gas dynamics, 1953, Proc. Roy. Soc., vol. 220, pp. 339-455.

25. G.B. Whitham, On the propagation of shock waves through regions of non-uniform area or flow, 1958, J. Fluid Mech., vol. 4, pp. 337–360.

26. G. Nath, J.P. Vishwakarma, V.K. Shrivastava, and A.K. Sinha., Propagation of magnetogasdynamic shock waves in a self-gravitating gas with exponentially varying density, 2013, J. Theor. Appl. Phys., vol. 7, p. 15. DOI: 10.1186/2251-7235-7-153.

27. J.P. Vishwakarma, M. Singh, Propagation of spherical shock waves through self-gravitating non-ideal gas with or without overtaking disturbances, 2013, Modelling, Measurment and Control B 82, 15-33.

28. K.K. Singh, Self-similar flow behind a cylindrical shock wave in a self-gravitating rotating gas with heat conduction and radiation heat flux, 2012, AMSE Journals, Modelling B, Vol. 81, pp. 61-81.

29. G. Nath, A.K. Sinha, Magnetogasdynamic shock waves in non-ideal gas under gravitational field-isothermal flow, 2017, Int. J. Appl. Comp. Math., vol. 3, pp. 225-238.

30. J.P.Vishwakarma, N. Patel, Magnetogasdynamic cylindrical shock waves in a rotating non-ideal gas with radiation heat flux, 2015, J. Eng. Phys. Thermophys., 88,521-530.

31. K.K. Singh, B. Nath, Similarity solutions for the flow behind an exponential shock in a rotating non-ideal gas with heat conduction and radiation heat fluxes, 2014, J. Eng. Phys. Thermophys, vol. 87, pp. 973-983.

32. S.I. Pai, Inviscid flow of radiation gasdynamics (High temperature inviscid flow of ideal radiating gas, analyzing effects of radiation pressure and energy on flow field), 1969, J. Math. Phys. Sci. 3, pp. 361-70.