Optimal classification and similarity solution for unsteady flow behind a shock wave in a perfectly conducting dusty gas with magnetic field using group invariance method

Abstract

We have studied the one dimensional unsteady isothermal and adiabatic flows behind the shock waves in a non-ideal dusty gas under the effect of magnetic field using Lie group invariance method. This method is used to get the infinitesimal generators of the Lie algebra. The concept of optimal systems is used for minimizing the group-invariant solutions. The optimal system of subalgebra is used to obtain the similarity variables and similarity transformation which transform the set of partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). Consequently, the system of ODEs in different cases is solved by using Runge–Kutta (R-K) method of forth order to derive the similarity solution in the cases of isothermal and adiabatic flows. The detailed discussions are illustrated by graph through numerical calculation in the case of power-law shock path. It is shown that the shock wave strength has decaying effects with an increase in the values of Alfven Mach number, adiabatic exponent of the gas, and gas non-idealness parameter. Also, the shock strength increases with an increment in the values of mass concentration of solid particles, magnetic field variation exponent, the ratio of density of solid particles to the gas initial density, and by changing the geometry from cylindrical to spherical.