Lie Scaling Symmetry Analysis and Exact Solution of Riemann Problem in Polytropic Perfect Gas With Magnetic Field

ABSTRACT The aim of the present paper is to study the Riemann problem based on a particular exact solution of a quasilinear hyperbolic system of partial differential equations (PDEs) obtained by using Lie scaling symmetry analysis. This system of PDEs governs one‐dimensional unsteady flow of an inviscid and polytropic perfect gas in planar geometry influenced by a transverse magnetic field. Based on the linearly degenerate and genuinely nonlinear nature of the characteristic fields, the solutions of the Riemann problem are classified into three types: Rarefaction waves, contact discontinuities, and shock waves. Further, we have constructed a particular solution in exact form for rarefaction waves, contact discontinuities, and shock waves. It is found that an increase in the shock Cowling number leads to a reduction in the jump of the flow variables, thereby resulting in smoother variations across the shock front. For the contact discontinuity, the density exhibits a finite jump across the second characteristic line, whereas the velocity and pressure remain continuous across this line. The methodology discussed in the present study to obtain the exact solution is very helpful for understanding and solving the Riemann problem associated with several physical phenomena.