Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics
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Delta Shock Wave Solution of the Riemann Problem for the Non-homogeneous Modified Chaplygin Gasdynamics Rahul Kumar Chaturvedi1 · L. P. Singh1 · Dia Zeidan2
Received: 27 January 2020 / Revised: 3 October 2020 / Accepted: 16 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The motivation of the present study is to derive the solution of the Riemann problem for modified Chaplygin gas equations in the presence of constant external force. The analysis leads to the fact that in some special circumstances delta shock appears in the solution of the Riemann problem. Also, the Rankine–Hugoniot relations for delta shock wave which are utilized to determine the strength, position and propagation speed of the delta shocks have been derived. Delta shock wave solution to the Riemann problem for the modified Chaplygin gas equation is obtained. It is found that the external force term, appearing in the governing equations, influences the Riemann solution for the modified Chaplygin gas equation. Keywords Riemann problem · Delta shocks · Modified Chaplygin gas · Rankine–Hugoniot conditions · Friction Mathematics Subject Classification 35L45 · 35L65 · 35L67 · 76N99
1 Introduction This paper investigates the classical and non-classical Riemann solution for the inhomogeneous system of partial differential equations which is of great interest to many researchers in applied mathematics and engineering sciences. The study of the hyperbolic systems have significant physical background which is interesting as it leads to diverse complex problems in Mathematics. For instance, authors in [1] have studied for the first time the homogeneous
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Dia Zeidan [email protected] Rahul Kumar Chaturvedi [email protected] L. P. Singh [email protected]
1
Department of Mathematical Sciences, Indian Institute of Technology, Banaras Hindu University, Varanasi 221005, India
2
School of Basic Sciences and Humanities, German Jordanian University, Amman, Jordan
123
Journal of Dynamics and Differential Equations
MCG equation model. In exotic background of fluid phenomenon, the MCG model plays an important role to describe the accelerated expansion of the universe and evolution of the perturbations of energy density. Also, It describes the dark energy and dark matter in the unified form. For further applications related to the modified Chaplygin gas equation, the interested reader is referred to [2,5,8,15,20] and references cited therein. In this paper, we propose to consider the following one-dimensional modified Chaplygin gas equation with constant external force term as, ∂t ρ + ∂x (ρv) = 0, (1) ∂t (ρv) + ∂x ρv 2 + P = ηρ, where ρ is the density and v is the velocity of the gas. The parameter η is a constant. Here the constant external force term appearing in the momentum equation of the model is treated as a coulomb-like friction term. The scalar P = P(ρ, α) is known as the modified Chaplygin gas pressure defined as P(ρ, α) = α p(ρ), where the parameter α > 0 is constant.
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