Steepening of waves in non-ideal reacting gas with dust particles
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ORIGINAL PAPER
Steepening of waves in non-ideal reacting gas with dust particles K Sharma, A Chauhan and R Arora* Department of Applied Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, India Received: 04 August 2019 / Accepted: 04 May 2020
Abstract: In this paper, the surface theory and compatibility conditions are used to describe the behavior of wave propagation and their culmination into a shock wave in a non-ideal reacting gas with dust particles. The one-dimensional steepening of waves is considered. A Bernoulli-type transport equation for the velocity gradient is obtained. A numerical approach is used to explain the effects of van der Waals excluded volume of the medium, radiation parameter, ratio of specific heats and dust particles. Keywords: Singular surface theory; Compatibility conditions; Dusty gas; Reacting gas; Breaking of waves
1. Introduction Study of propagation of nonlinear waves has become a very interesting topic for both the mathematicians and the physicists because of their applications in geophysics, nuclear science, fusion reaction, astrophysics and in the motion of satellites, etc. When the nonlinear waves propagate in a medium, certain kinds of discontinuities are encountered, known as shock waves or acceleration waves. Shock waves also play an important role in the medical field. The wave is considered as moving surface under which the variables and their derivatives undergo certain kinds of discontinuities that are carried along by the surface. These discontinuities are bound to be interrelated. The relations connecting the field variables, or their derivatives on the two sides of the discontinuity surface, are known as compatibility conditions and arise due to the dynamical conditions that determine the behavior of the material medium. The first set of compatibility conditions, namely Rankine–Hugoniot jump conditions, is a consequence of the conservation laws that hold across the discontinuity surface. Furthermore, those relation which connects the first-order derivatives of the field variables on the two sides of the discontinuity surface with its speed of propagation is known as compatibility condition of first order. The compatibility conditions for the second- and the higher-order derivatives can be obtained with the assumption of a smooth wavefront. The geometrical and
kinematical compatibility conditions of first and second order were developed by Thomas [1]. The growth and decay of the discontinuity in fluid were investigated by Coleman and Gurtin [2]. Many researchers have investigated the behavior of shock waves in fluids using the singular surface theory, and their research work can be seen from ([3–8]). In 1981, Ying et al. [9] showed the relation between the structure of the solution of reacting gas and that of Riemann problem. The Riemann problem for reacting gas has also been studied by Chorin et al. [10]. Barenblatt et al. [11] investigated the dependency of the solution of the problem on the radius of the corresponding pipe, from where the ga
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