Similarity solutions of converging shock waves in an ideal relaxing gas with dust particles
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Similarity solutions of converging shock waves in an ideal relaxing gas with dust particles Swati Chauhana , Antim Chauhanb , Rajan Arorac Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, India Received: 7 July 2020 / Accepted: 1 October 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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.
1 Introduction In various fields, the evolutionary behavior of shock waves draws attention from both the mathematicians and physicists due to their large applications. The study of shock waves in a mixture of gases with dust particles has become important due to their various applications in environmental and industrial fields such as lunar ash flow, nozzle flow, black hole theory, volcanic explosion, nuclear blasts, supersonic flight in dusty air, the formation of dusty crystals and many other diverse phenomena (see [1–4]). There is huge literature which concerns with the study of propagation of shock waves in dusty gas medium [5,6]. The shock waves occur due to the collision of particles, in which the energy is dissipated rapidly with distance. These situations may occur during transportation of particles in mine galleries, grain mills and chemical plants. Since the strong shock waves produce high temperature and pressure at
a e-mail: [email protected] b e-mail: [email protected] c e-mails: [email protected]; [email protected] (corresponding author)
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the center of convergence, it becomes a field of continuing research interest. Also, due to this pro
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