Lie group of similarity analysis of shock waves in viscous flow under magnetic field

Abstract

This paper investigates the propagation of planar shock waves in an ideal gas under the viscous stress and magnetic field using the Lie group of similarity analysis. The ambient density and magnetic field both vary with shock radius as power and exponential law. Newton’s law of viscosity has been used and shock jump conditions have been derived for viscous flow by introducing shock Reynolds number $R{e}_s$. The system of governing partial differential equations is reduced into a system of ordinary differential equations using the Lie group of invariance method and numerical solutions have been obtained for power and exponential law both. The effects of shock Reynolds number $R{e}_s=10$ (highly viscous flow), 50, 200, 1000 (slightly viscous flow), and $R{e}_s\to \infty$ (non-viscous flow), Alfven-Mach number, and ratio of specific heats have been discussed on the shock strength, piston position and flow variables behind the shock front. The magnetic field enhances the effect of viscous stress in exponential law but reduces in power law. All flow variables except viscous stress increase for power law and decrease for exponential law in viscous flow in comparison to non-viscous flow. Comparison of results of highly non-viscous flow with the corresponding results of inviscid flow establishes the validity of the model presented in this work. The results of this paper show the significant effect of viscosity on shock propagation contrary to the negligible effect in earlier studies.