Similarity solutions using the optimal system of solvable Lie subalgebras for shock waves in a perfectly conducting rotating real dusty gas with magnetic field by applying the Lie symmetry method

Abstract The present article is concerned with the study of cylindrical shock wave in a perfectly conducting non-ideal dusty gas influenced by magnetic field for isothermal flow in the rotating medium. The present phenomenon is governed by the hyperbolic system of non-linear partial differential equations (PDEs). To evaluate safety and predict disasters caused by explosions, it is essential to thoroughly understand the behavior of solutions to such a hyperbolic system of non-linear PDEs. To investigate all possible solutions, we have used the Lie symmetry method. Initially, we have constructed the optimal system of one-dimensional solvable Lie subalgebras. This optimal system provides vital information about group-invariant solutions, while our aim is to analyze similarity solution. The similarity solutions exist in three cases; one with exponential law shock path and two with power law shock path. Consequently, it is found that the shock wave decays with an increment in the physical parameters such as gas non-idealness, mass concentration of solid particles in the mixture, adiabatic exponent, and shock Cowling number. Also, the shock wave enhanced with an increment in the ratio of density of solid particles to initial species density of the gas. Moreover, it is found that the shock wave decays due to change in magnetic field from axial to azimuthal. The shock wave is more powerful in exponential law case in comparison to power law case.