Similarity Solution of Spherical Shock Waves -Effect of Viscosity
DOI:
https://doi.org/10.4067/S0716-09172016000100002Keywords:
Similarity solutions, shock waves, magnetogasdynamics, viscosity, Rankine-Hugoniot condition, Mie-Gruneisen type, numerical solution, soluciones de similaridad, ondas de choque, magnetodinámica, viscosidad, condición de Rankine-Hugoniot.Abstract
In this paper we investigated self-similar Solutions for Magneto Hy-drodynamic shock waves for the equation of state of Mie-Gruneisen type. Solutions are obtained numerically and the effect of viscosity (K) and the non-idealness parameter (d) on the self-similar solutions are studied in detail. The findings confirmed that, the non-idealness parameter and the viscosity parameter have major effect on the shock strength and the flow variables. All discontinuities of the physical pa-rameters are removed by the viscosity and complete flow field depends upon the magnitude of the viscosity. The obtained results are in good agreement with the results obtained by some of the researchers. All the analysis is presented pictorially in this paper.References
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[28] K. Zumbrun, The refined inviscid stability condition and cellular instability of viscous shock waves, Physica D: Nonlinear Phenomena., 239? (13), pp. 1180-1187, (2010).?
[2] I. Ballai, E. Forgacs-Dajka, A. Marcu, Dispersive shock waves in the? solar wind, Astronomische Nachrichten, 328 (8), pp. 734-737, (2007).?
[3] J. Blazek, Computational Fluid Dynamics: Principles and Applications, Elsevier, Amsterdam, (2001).?
[4] E. J. Caramana, M. J. Shashkov, P. P. Whalen, Formulations of artificial viscosity for multi-dimensional shock wave computations, J. Comput. Phys., 144 (1), pp. 70-97, (1998).?
[5] A. V. Chikitkin, B. V. Rogov, G. A. Tirsky, S. V. Utyuzhnikov, Effect? of bulk viscosity in supersonic flow past spacecraft, Applied Numerical? Mathematics, doi: 10.1016/j.apnum.2014.01.004 (2014).?
[6] R.F. Chisnell, An analytic description of converging shock waves, J.? Fluid. Mech., 354, pp. 357-375, (1998).?
[7] V. Genot, Analytical solutions for anisotropic MHD shocks, Astro.? Phys. Space Sci. Trans., 5, pp. 31-34, (2009).?
[8] W. Gretler, R. Regenfelder, Similarity solution for laser-driven shock? waves in a particle-laden gas, Fluid Dynamics Research, 28 (5), pp.? 369-382, (2001).?
[9] P. Hafner, Strong convergent shock waves near the center of convergence: A power series solution, SIAM J. Appl. Math., 48 (6), pp. 1244-1261, (1998).
[10] P. M. Jordan, M. R. Meyer, A. Puri, Causal implications of viscous? damping in compressible fluid flows, Phys. Rev., E. 62, pp. 7918 - 7926, (2000).
[11] V. Khodadad, N. Khazraiyan, Numerical modeling of ballistic penetration of long rods into ceramic/metal armors,8th International
LS-DYNA Users Conference-Drop/Impact Simulations, 14, pp. 39-50,
(2004).
[12] V. P. Korobeinikov, Problems in the theory of point explosion in gases,? Providence, American Mathematical Society, Rhode Island (1976).?
[13] N. P. Korzhov, V. V. Mishin, V.M. Tomozov, On the viscous interaction of solar wind streams, Soviet Astronomy, 29, pp. 215-218, (1985).?
[14] L.D. Landau, E. M. Lifshitz, Fluid Mechanics, Pergamon press, New? York (1987).?
[15] R. Lazarus, R. D. Richtmyer, Similarity solutions for converging? shocks, Technical report LA- 6823-MS. Los Alamos Scientific Laboratory (1977).?
[16] W. H. Lee, P.P. Whalen, Calculation of shock problems by using four? different schemes, International Conference on Numerical Methods for? Transient and Coupled Problems. Venice, Italy (1984).?
[17] A. A. Maslov, S. G.Mironov, A. N. Kudryavtsev, T. V. Poplavskaya,? I.S. Tsyryulnikov, Wave process in a viscous shock layer and control? of fluctuations, J. Fluid Mech., 650, pp. 81-118, (2010).?
[18] J. A. Orta, M.A. Huerta, G. C. Boynton, Magnetohydrodynamic shock? heating of the solar corona, The Astrophysical Journal., 596, pp. 646-655, (2003).
[19] A. Ramu, M. P. Ranga Rao, Converging spherical and cylindrical shock? waves, J. Eng. Math. 27 (4), pp. 411-417, (1993).?
[20] A. Ramu, N. Dunna, D. K. Satpathi, Numerical study of shock waves? in non-idea magnetogasdynamics, Journal of Egyptian Mathematical? Society, 24 (1), pp. 116-124 (2016).?
[21] L. Richard, Similarity solutions for a spherical shock wave, J. Appl.? Phys., 26 (8), pp. 954-960, (1955).?
[22] V. D. Sharma, R. Arora, Similarity solutions for strong shocks in an? ideal gas, Stud. Appl. Math. 114 (4), pp. 375-394, (2005).?
[23] K. P. Stanyukovich, Unsteady motion of Continuous Medi,. Pergamon? Press, New York (1960).?
[24] G. I. Taylor, The formation of a blast wave by a very intense explosionI and II: The atomic explosion of 1945, Proc. Roy. Soc., London, 201 (1065), pp. 159-186, (1950).
[25] J. Von Neumann, R.D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys. , 21 (3), pp. 232-237,? (1950).?
[26] G. B.Whitham, Linear and Non-linear Waves. John Wiley & Sons,? New York, (1974).?
[27] Ya. B.Zeldovich, Y.P. Raizer, Physics of Shock waves and HighTemperature Hydrodynamic Phenomenon. Vol. II., Academic Press,? New York, (1966).?
[28] K. Zumbrun, The refined inviscid stability condition and cellular instability of viscous shock waves, Physica D: Nonlinear Phenomena., 239? (13), pp. 1180-1187, (2010).?
Published
2017-03-23
How to Cite
[1]
N. Dunna, A. Ramu, and D. K. Satpathi, “Similarity Solution of Spherical Shock Waves -Effect of Viscosity”, Proyecciones (Antofagasta, On line), vol. 35, no. 1, pp. 11-31, Mar. 2017.
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