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Shock Wave Driven Out by a Piston in a Mixture of a Non-ideal Gas and Small Solid Particles Under the Influence of Azimuthal or Axial Magnetic Field

  • General and Applied Physics
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Abstract

One-dimensional self-similar unsteady flows behind a spherical or cylindrical shock wave driven out by a piston moving with time according to a power law in a dusty gas is investigated. The medium is assumed to be the mixture of small solid particles of micro-size and a non-ideal gas. The solid particles are uniformly distributed in the mixture, and the shock wave is assumed to be driven by the inner surface (piston). It is assumed that the equilibrium flow-conditions are maintained and the moving piston continuously supplies the variable energy input. Similarity solutions exist only when the surrounding medium is of constant density. Solutions are obtained, in both cases, when the flow between the shock and the piston is isothermal or adiabatic. The shock waves in non-ideal dusty gas can be important for the description of shocks in supernova explosions, in the study of a flare produced shock in the solar wind, the central part of starburst galaxies, nuclear explosion, rupture of the pressurized vessel, in the analysis of data from exploding wire experiments, and cylindrically symmetric hypersonic flow problems associated with meteors or re-entry vehicles, etc. The findings of the present work provided a clear picture of whether and how the non-idealness of the gas and the presence of the magnetic field affect the propagation of shock and the flow behind it. It is interesting to note that in the presence of azimuthal magnetic field, the pressure and density vanish at the piston and hence a vacuum is formed at the center of symmetry for both the isothermal and adiabatic flows, which is in excellent agreement with the laboratory condition to produce the shock wave.

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Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. F. Robben, L. Talbot, Measurement of shock wave thickness by the electron beam fluorescence method. Phys.Fluids. 9(4), 633–643 (1966)

    ADS  Google Scholar 

  2. G. Jagadeesh, Fascinating world of shock waves. Resonance. 13(8), 752–767 (2008)

    Google Scholar 

  3. F. Sharipov, F.C. Dias, Temperature dependence of shock wave structure in helium and neon. Physics of Fluids. 31(3), 037109 (2019)

    ADS  Google Scholar 

  4. D.L. Frost, Heterogeneous particle-laden blast waves. Shock Waves. 28, 439–449 (2018)

    ADS  Google Scholar 

  5. A.R. Miles, The blast-wave-driven instability as a vehicle for understanding supernova explosion structure. Astrophys. J. 696(1), 498 (2009)

    ADS  Google Scholar 

  6. T. Inoue, R. Yamazaki, S.I. Inutsuka, Turbulence and magnetic field amplification in supernova remnants: interactions between a strong shock wave and multiphase interstellar medium. Astrophys. J. 695(2), 825 (2009)

    ADS  Google Scholar 

  7. S.W. Kieffer, Blast dynamics at mount St Helens on 18 May 1980. Nature. 291(5816), 568 (1981)

    ADS  Google Scholar 

  8. Y. Formenti, T.H. Druitt, K. Kelfoun, Characterisation of the 1997 Vulcanian explosions of soufrière Hills Volcano, Montserrat, by video analysis. Bull. Volcanol. 65(8), 587–605 (2003)

    ADS  Google Scholar 

  9. Y. Aglitskiy, A.L. Velikovich, M. Karasik, N. Metzler, S.T. Zalesak, A.J. Schmitt, S.P. Obenschain, Basic hydrodynamics of Richtmyer–Meshkov-type growth and oscillations in the inertial confinement fusion-relevant conditions Philosophical Transactions of the Royal Society a: Mathematical. Phys. Eng. Sci. 368(1916), 1739–1768 (2010)

    Google Scholar 

  10. R.K. Eckhoff, Dust explosion prevention and mitigation, status and developments in basic knowledge and in practical application (2009)

  11. R.M. Allen, D.J. Kirkpatrick, A.W. Longbottom, A.M. Milne, N.K. Bourne, in Experimental and Numerical Study of Free-Field Blast Mitigation. AIP Conference Proceedings. AIP, Vol. 706, (2004), pp. 823–826

  12. D.L. Frost, C. Ornthanalai, Z. Zarei, V. Tanguay, F. Zhang, Particle momentum effects from the detonation of heterogeneous explosives. J. Appl. Phys. 101(11), 113529 (2007)

    ADS  Google Scholar 

  13. G. Nath, P.K. Sahu, Flow behind an exponential shock wave in a rotational axisymmetric perfect gas with magnetic field and variable density. SpringerPlus. 5(1), 1509 (2016)

    Google Scholar 

  14. G. Nath, P.K. Sahu, Similarity solution for the flow behind a cylindrical shock wave in a rotational axisymmetric gas with magnetic field and monochromatic radiation. Ain Shams Eng. J. 9(4), 1151–1159 (2018)

    Google Scholar 

  15. S.I. Anisimov, O.M. Spiner, Motion of an almost ideal gas in the presence of a strong point explosion: PMM. J. Appl. Math. Mech. 36(5), 883–887 (1972)

    Google Scholar 

  16. M.P.R. Rao, N.K. Purohit, Self-similar piston problem in non-ideal gas. Int. J. Eng. Sci. 14(1), 91–97 (1976)

    MATH  Google Scholar 

  17. C.C. Wu, P.H. Roberts, Shock-wave propagation in a sonoluminescing gas bubble. Phys. rev. lett. 70(22), 3424 (1993)

    ADS  Google Scholar 

  18. P.H. Roberts, C.C. Wu, Structure and stability of a spherical implosion. Phys. Lett. A. 213(1-2), 59–64 (1996)

    ADS  Google Scholar 

  19. G. Nath, P.K. Sahu, Flow behind an exponential shock wave in a rotational axisymmetric non-ideal gas with conduction and radiation heat flux. International Journal of Applied and Computational Mathematics. 3(4), 2785–2801 (2017)

    MathSciNet  MATH  Google Scholar 

  20. P.K. Sahu, Similarity solution for a spherical shock wave in a non-ideal gas under the influence of gravitational field and monochromatic radiation with increasing energy (2019)

  21. P.K. Sahu, in Similarity Solution for the Flow Behind an Exponential Shock Wave in a Rotational Axisymmetric Non-ideal Gas Under the Influence of Gravitational Field with Conductive and Radiative Heat Fluxes. Dawn S., Balas V., Esposito A., Gope S. (eds) Intelligent Techniques and Applications in Science and Technology. ICIMSAT 2019. Learning and Analytics in Intelligent Systems, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-030-42363-6_122https://doi.org/10.1007/978-3-030-42363-6_122, (2020)

  22. N. Zhao, A. Mentrelli, T. Ruggeri, M. Sugiyama, Admissible shock waves and shock-induced phase transitions in a van der Waals fluid. Physics of fluids. 23(8), 086101 (2011)

    ADS  MATH  Google Scholar 

  23. S.I. Pai, S. Menon, Z.Q. Fan, Similarity solutions of a strong shock wave propagation in a mixture of a gas and dusty particles. Int. J. Eng. Sci. 18(12), 1365–1373 (1980)

    MATH  Google Scholar 

  24. F. Higashino, T. Suzuki, The effect of particles on blast waves in a dusty gas. Zeitschrift fü,r Naturforschung A. 35(12), 1330–1336 (1980)

    ADS  Google Scholar 

  25. H. Miura, I.I. Glass, in Development of the flow induced by a piston moving impulsively in a dusty gas. Proceedings of the Royal Society of London A. Mathematical and Physical Sciences, Vol. 397, (1985), pp. 295–309

  26. F. Conforto, Wave features and group analysis for an axi-symmetric model of a dusty gas. International journal of non-linear mechanics. 35(5), 925–930 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  27. O. Igra, G. Hu, J. Falcovitz, B.Y. Wang, Shock wave reflection from a wedge in a dusty gas. International journal of multiphase flow. 30(9), 1139–1169 (2004)

    MATH  Google Scholar 

  28. W. Gretler, R. Regenfelder, Strong shock waves generated by a piston moving in a dust-laden gas under isothermal condition. European Journal of Mechanics-B/Fluids. 24(2), 205–218 (2005)

    ADS  MATH  Google Scholar 

  29. S.I. Popel, A.A. Gisko, Charged dust and shock phenomena in the solar system. Nonlinear Processes Geophys. 13(2), 223–229 (2006)

    ADS  Google Scholar 

  30. J.P. Vishwakarma, G. Nath, A self-similar solution of a shock propagation in a mixture of a non-ideal gas and small solid particles. Meccanica. 44(3), 239 (2009)

    MATH  Google Scholar 

  31. S.I. Pai, Two-phase flows Springer-Verlag 3 (2013)

  32. G. Nath, P.K. Sahu, Self-similar solution of a cylindrical shock wave under the action of monochromatic radiation in a rotational axisymmetric dusty gas. Commun. Theor. Phys. 67(3), 327 (2017)

    ADS  Google Scholar 

  33. G. Nath, P.K. Sahu, Propagation of a cylindrical shock wave in a mixture of a non-ideal gas and small solid particles under the action of monochromatic radiation. Combustion, Explosion, and Shock Waves. 53(3), 298–308 (2017)

    Google Scholar 

  34. P.K. Sahu, Cylindrical shock waves in rotational axisymmetric non-ideal dusty gas with increasing energy under the action of monochromatic radiation. Physics of Fluids. 29(8), 086102 (2017)

    ADS  Google Scholar 

  35. P.K. Sahu, Self-similar solution of spherical shock wave propagation in a mixture of a gas and small solid particles with increasing energy under the influence of gravitational field and monochromatic radiation. Commun. Theor. Phys. 70(2), 197 (2018)

    ADS  MathSciNet  Google Scholar 

  36. J.P. Vishwakarma, P. Lata, Self-similar solution of a shock wave propagation in a perfectly conducting dusty gas. International Journal of Research in Advent Technology. 6(7), 1789–1800 (2018)

    Google Scholar 

  37. K. Shibasaki, S. Shibasaki, G. Jagadeesh, M. Sun, K. Takayama, in Development of a high-speed cylindrical rotor device for industrial applications of shock waves. Proceedings of 23rd International Symposium on Shock Waves, (2001)

  38. J.S. Park, S.W. Baek, Interaction of a moving shock wave with a two-phase reacting medium. International journal of heat and mass transfer. 46(24), 4717–4732 (2003)

    MATH  Google Scholar 

  39. N.N. Smirnov, V.F. Nikitin, Modeling and simulation of hydrogen combustion in engines. Int. J. Hydrogen Energy. 39(2), 1122–1136 (2014)

    Google Scholar 

  40. N.N. Smirnov, V.F. Nikitin, V.R. Dushin, Y.G. Filippov, V.A. Nerchenko, J. Khadem, Combustion onset in non-uniform dispersed mixtures. Acta Astronaut. 115, 94–101 (2015)

    Google Scholar 

  41. L. Hartmann. Accretion Processes in Star Formation (Cambridge University Press, Cambridge, 1998)

    Google Scholar 

  42. B. Balick, A. Frank, Shapes and shaping of planetary nebulae. Annu. Rev. Astron. Astrophys. 40, 439 (2002)

    ADS  Google Scholar 

  43. P.K. Sahu, in Unsteady Flow Behind an MHD Exponential Shock Wave in a Rotational Axisymmetric Non-ideal Gas with Conductive and Radiative Heat Fluxes. Dawn S., Balas V., Esposito A., Gope S. (eds) Intelligent Techniques and Applications in Science and Technology. ICIMSAT 2019. Learning and Analytics in Intelligent Systems, vol 12. Springer, Cham doi=https://doi.org/10.1007/978-3-030-42363-6_121, (2020)

  44. P.K. Sahu, Propagation of an exponential shock wave in a rotational axisymmetric isothermal or adiabatic flow of a self-gravitating non-ideal gas under the influence of axial or azimuthal magnetic field. Chaos, Solitons & Fractals. 135, 109739 (2020)

    ADS  MathSciNet  Google Scholar 

  45. M.K. Verma, Statistical theory of magnetohydrodynamic turbulence: recent results. Phys. Rep. 401(5-6), 229–380 (2004)

    ADS  MathSciNet  Google Scholar 

  46. Y.B. Zel’Dovich, Y.P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena Academic Press (1967)

  47. H. Steiner, T. Hirschler, A self-similar solution of a shock propagation in a dusty gas. European Journal of Mechanics-B/Fluids. 21(3), 371–380 (2002)

    ADS  MathSciNet  MATH  Google Scholar 

  48. L.I. Sedov, Similarity and dimensional methods in mechanics.Academic Press New York (1959)

  49. R.A. Freeman, J.D. Craggs, Shock waves from spark discharges. J. Phys. D: Appl. Phys. 2, 421 (1969)

    ADS  Google Scholar 

  50. E.A. Moelwyn-Hughes. Physical Chemistry (Pergamon Press, London, 1961)

    Google Scholar 

  51. A.V. Fedorov, Y.V. Kratova, Interaction of heterogeneous detonation waves with the cloud of inert particles. Heat Transfer Research. 43(2), 123 (2012)

    Google Scholar 

  52. M. Onsi, H. Przysiezniak, J.M. Pearson, Equation of state of homogeneous nuclear matter and the symmetry coefficient. Physical Review C. 50(1), 460 (1994)

    ADS  Google Scholar 

  53. R.H. Casali, D.P. Menezes, Adiabatic index of hot and cold compact objects. Braz. J. Phys. 40(2), 166–171 (2010)

    Google Scholar 

  54. P. Rosenau, S. Frankenthal, Equatorial propagation of axisymmetric magnetohydrodynamic shocks. The Physics of Fluids. 19(12), 1889–1899 (1976)

    ADS  MATH  Google Scholar 

  55. I.F. Barna, L. Matyas, Analytic solutions for the one-dimensional compressible Euler equation with heat conduction and with different kind of equations of state. Miskolc Mathematical Notes. 14(3), 785–799 (2013)

    MathSciNet  MATH  Google Scholar 

  56. I.F. Barna, L. Mátyás, Analytic self-similar solutions of the Oberbeck–Boussinesq equations. Chaos, Solitons and Fractals. 78, 249–255 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  57. O. Igra, J. Falcovitz, H. Reichenbach, W. Heilig, Experimental and numerical study of the interaction between a planar shock wave and a square cavity. J. Fluid Mech. 313, 105–130 (1996)

    ADS  Google Scholar 

  58. O. Igra, X. Wu, J. Falcovitz, T. Meguro, K. Takayama, W. Heilig, Experimental and theoretical study of shock wave propagation through double-bend ducts. J. Fluid Mech. 437, 255–282 (2001)

    ADS  MATH  Google Scholar 

  59. J. Falcovitz, G. Alfandary, G. Ben-Dor, Numerical simulation of the head-on reflection of a regular reflection. International journal for numerical methods in fluids. 17(12), 1055–1077 (1993)

    ADS  MATH  Google Scholar 

  60. J. Falcovitz, M. Ben-Artzi, Recent developments of the GRP method. JSME International Journal Series B Fluids and Thermal Engineering. 38(4), 497–517 (1995)

    ADS  Google Scholar 

  61. G.J. Hutchens, Approximate cylindrical blast theory: near-field solutions. Journal of applied physics. 77(7), 2912–2915 (1995)

    ADS  Google Scholar 

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Acknowledgments

The author is thankful to Prof. M. K. Verma, Department of Physics, Indian Institute of Technology Kanpur, Kanpur–208016, India for valuable suggestions and fruitful discussions.

Funding

This research was supported by the research grant no. TAR/2018/000150 under Teachers Associateship for Research Excellence (TARE) scheme from the Science and Engineering Research Board (SERB), India. The author gratefully acknowledges the financial support of SERB.

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  1. Department of Mathematics, Government Shyama Prasad Mukharjee College Sitapur, Sitapur, Chhattisgarh, 497111, India

    P. K. Sahu

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Sahu, P.K. Shock Wave Driven Out by a Piston in a Mixture of a Non-ideal Gas and Small Solid Particles Under the Influence of Azimuthal or Axial Magnetic Field. Braz J Phys 50, 548–565 (2020). https://doi.org/10.1007/s13538-020-00762-x

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