Abstract
The present paper demonstrates the analysis of converging shock wave in an ideal relaxing gas with dust particles via Lie group theoretic method. The Lie group of transformations is used to determine the whole range of self-similar solutions to a problem involving both planar and non-planar flows in an ideal, relaxing and dusty gas involving strong converging shocks. For the strong shock waves, all the necessary group invariance properties associated with ambient gas are presented and the general form of rate of relaxation for which the self-similar solutions exist is determined. By using the invariance surface conditions, we determine the infinitesimal generators of Lie group of transformations associated with the system of partial differential equations and on the basis of arbitrary constants occurring in the expressions for the generators, four different possible cases involving self-similar solutions are reckoned. For the different values of dust parameters, the similarity exponents are obtained numerically and comparison is made with the similarity exponents obtained from the characteristic rule (or CCW method). The effects of mass fraction of dust particle, relative specific heat and ratio of density of dust particle to density of gas, on the flow variables and shock formation, have been shown. The patterns of all flow variables behind the shock are analyzed graphically.
Similar content being viewed by others
References
E.F. Medici, J.S. Allen, G.P. Waite, Modeling shock waves generated by explosive volcanic eruptions. Geophys. Res. Lett. 41, 414–421 (2014)
Y. BenTov, J. Swearngin, Gravitational shock waves on rotating black holes. General Relativ. Gravitat. 51, 1–41 (2019)
R.S. Iyenger, Shock wave propagation from a nuclear blast. Nature 203, 746–747 (1964)
R.E. Pudritz, N.K.R. Kevlahan, Shock interactions, turbulence and the origin of the stellar mass spectrum. Philos. Trans. R. Soc. A 371, 1–16 (2013)
J. Yin, J. Ding, X. Luo, Numerical study on dusty shock reflection over a double wedge. Phys. Fluids 30(1), 013304 (2018)
M. Chadha, J. Jena, Self-similar solutions and converging shocks in a non-ideal gas with dust particles. Int. J. Non-Linear Mech. 65, 164–172 (2014)
K. Takayama, T. Saito, Shock wave/geophysical and medical applications. Annu. Rev. Fluid Mech. 36, 347–379 (2014)
G. Guderley, Starke kugelige und zylindrische verdichtungsstosse in der nahe des kugelmittelpunktes bzw der zylinderachse. zuftfahrtforschung 19, 302–312 (1942)
Y.B. Zeldovich, Y.P. Raizer, Physics of Shock Waves and High Temperature Hydrodynamic Phenomena (Academic Press, New York, NY, II, 1967)
P. Hafner, Strong convergent shock waves near the center of convergence: a power series solution. SIAM J. Appl. Math. 48(6), 1244–1261 (1988)
R.B. Lazarus, Self-similar solutions for converging shocks and collapsing cavities. SIAM J. Numer. Anal. 18, 316–371 (1981)
R.F. Chisnell, An analytic description of converging shock waves. J. Fluid Mech. 354, 357–375 (1998)
M. Vandyke, A.J. Guttmann, The converging shock wave from a spherical or cylindrical piston. J. Fluid Mech. 120, 451–462 (1982)
R. Arora, V.D. Sharma, Convergence of strong shock in a Van Der Waals gas. SIAM J. Appl. Math. 66(5), 1825–1837 (2006)
A. Chauhan, R. Arora, A. Tomar, Convergence of strong shock waves in a non-ideal magnetogasdynamics. Phys. Fluids 30, 116105 (2018)
A. Tomar, R. Arora, A. Chauhan, Propagation of strong shock waves in a non-ideal gas. Acta Astronaut. 159, 96–104 (2019)
Z.M. Boyd, S.D. Ramsey, R.S. Baty, On the existence of self-similar converging shocks for arbitrary equation of state. Q. J. Mech. Appl. Math. 70, 401–417 (2017)
N. Ponchaut, H.G. Hornung, D.I. Pullin, C.A. Mouton, On imploding cylindrical and spherical shock waves in a perfect gas. J. Fluid Mech. 560, 103–122 (2006)
A. Chauhan, R. Arora, A. Tomar, Converging strong shock wave in magnetogasdynamics under isothermal condition. Ricerche di Mathematica 1–17 (2020)
N. Zhao, A. Mentrelli, T. Ruggeri, M. Sugiyama, Admissible shock waves and shock induced phase transitions in a Van der Waals fluid. Phys. Fluids 23, 86–101 (2011)
R. Arora, A. Tomar, V.P. Singh, Similarity solutions for strong shocks in a non-ideal gas. Math. Model. Anal. 17, 351–365 (2012)
R. Singh, J. Jena, One dimensional steepening of waves in non-ideal relaxing gas. Int. J. Nonlinear Mech. 77, 158–161 (2015)
V.D. Sharma, Ch. Radha, Similarity solutions for converging shocks in relaxing gas. Int. J. Eng. Sci. 33(4), 535–553 (1995)
S.A.V. Manickam, Ch. Radha, V.D. Sharma, Far field behaviour of waves in a vibrationally relaxing gas. Appl. Numer. Math. 45, 293–307 (2003)
J. Jena, V.D. Sharma, Interaction of a characteristic shock with a weak discontinuity in a relaxing gas. J. Eng. Math. 60, 43–53 (2008)
R. Arora, M.J. Siddiqui, V.P. Singh, Similarity method for imploding strong shocks in a non-ideal relaxing gas. Int. J. Non-Linear Mech. 57, 1–9 (2013)
S. Mehla, J. Jena, Shock wave kinematics in a relaxing gas with dust particles. Zeitschrift für Naturforschung A 74, 787–798 (2019)
G.I. Taylor, The formation of a blast wave by a very intense explosion I. Theoretical discussion. Proc. R. Soc. Lond. A 201, 159–174 (1950)
L.I. Sedov, Similarity and Dimensional Methods in Mechanics (Academic Press, New York, 1959)
P.J. Olver, Applications of Lie Groups to Differential Equations (Springer, New York, 1993)
G. Bluman, A. Cheviakov, Applications of Symmetry Methods to Partial Differential Equations (Springer, New York, 2010)
G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations (Springer, New York, 1974)
M.N. Ali, A.R. Seadawy, S.M. Husnine, Lie point symmetries exact solutions and conservation laws of perturbed Zakharov–Kuznetsov equation with higher-order dispersion term. Mod. Phys. Lett. A 34, 1950027 (2019)
M.N. Ali, A.R. Seadawy, S.M. Husnine, Lie point symmetries, conservation laws and exact solutions of (1+n)-dimensional modified Zakharov–Kuznetsov equation describing the waves in plasma physics. Pramana 91, 1–9 (2018)
G.B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974)
J.D. Logan, J.D.J. Perez, Similarity solutions for reactive shock hydrodynamics. SIAM J. Appl. Math. 39, 512–527 (1980)
Acknowledgements
The first author Swati Chauhan acknowledges the financial support from the “Ministry of Human Resource Development, New Delhi.” The second author Antim Chauhan acknowledges the research support from the “University Grant Commission (Govt. of India)” (Sr. No. 2121541039 with Ref No. 20/12/2015 (ii) EU-V).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chauhan, S., Chauhan, A. & Arora, R. Similarity solutions of converging shock waves in an ideal relaxing gas with dust particles. Eur. Phys. J. Plus 135, 825 (2020). https://doi.org/10.1140/epjp/s13360-020-00823-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/s13360-020-00823-9