Eur. Phys. J. Plus (2020) 135: 825
https://doi.org/10.1140/epjp/s13360-020-00823-9

Regular Article

Similarity solutions of converging shock waves in an ideal relaxing gas with dust particles

Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, India

^c rajan.arora@as.iitr.ac.in, rajana100@gmail.com

Received: 7 July 2020
Accepted: 1 October 2020
Published online: 14 October 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.