Evolution of $$C^{1}$$ C 1 -wave and its collision with the blast wave in one-dimensional non-ideal gas dynamics
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Evolution of C 1 -wave and its collision with the blast wave in one-dimensional non-ideal gas dynamics Rahul Kumar Chaturvedi1 L. P. Singh1
· Pooja Gupta1 · Shobhit Kumar Srivastava1 ·
Received: 5 November 2019 / Revised: 7 July 2020 / Accepted: 7 August 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract The aim of the present paper is to analyze the collision of discontinuity wave with blast wave in non-ideal gas flow through Lie group approach. The particular solution of the system of quasilinear hyperbolic PDEs, governing one-dimensional unsteady cylindrically symmetric motion in non-ideal gas, is obtained. The equation governing the evolutionary process of weak discontinuity in non-ideal gas is derived which is utilized to discuss the growth and decay process of the weak discontinuity. Furthermore, the collision of discontinuity wave with the blast wave is examined inside the state described by the exact solution of the blast wave. Also, the existence and uniqueness properties of reflection and transmission coefficients of the waves along with the discontinuity in the shock acceleration are discussed. Keywords Group-theoretic method · Wave interaction · Non-ideal gas · Blast wave Mathematics Subject Classification 76M60 · 76N15 · 76N30 · 76L05
1 Introduction We study the problem of collision of the waves within the context of the system of quasilinear hyperbolic PDEs. An interesting remark in the context of such system of PDEs is that its solution admits discontinuity waves. Discontinuity waves are the special class of the solution of the non-linear hyperbolic system of PDEs which are characterized by a discontinuity in the flow variable and its normal derivative. For a non-linear system, to find analytical solution and discussion of its physics in gasdynamics is of great interest from both physical and mathematical point of view. One of the most interesting problems on the theory of non-linear waves in gasdynamics is the process of the formation of shock waves. Detailed investigation towards a better understanding of the wave interaction phenomenon has been
Communicated by Abdellah Hadjadj.
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Rahul Kumar Chaturvedi [email protected] Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India 0123456789().: V,-vol
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studied by the authors Brun (1975), Morro (1978, 1980) and Jeffrey (1976). The authors Ruggeri (1980), Virgopia and Ferraioli (1994), Radha et al. (1993), Pandey and Sharma (2007), and Pandey (2010) have discussed the process of the interaction of waves in various fields like magnetogasdynamics, non-ideal gas, shallow water, elastic solids, relaxing gas, etc. Many authors Virgopia and Ferraioli (1994), Chaturvedi et al. (2019) studied the evolution of weak discontinuity in different material media. In past decades, Pandey et al. (2008), Arora et al. (2013), and Bira et al. (2016) have applied the group-theoretic method to obtai
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