Solutions for converging cylindrical and spherical shock waves in condensed matter equation of state

Abstract

A theoretical model for converging cylindrical and spherical shock waves in non-ideal gas characterized by condensed matter equation of state (EOS) of Mie-Gruneisen type is investigated. The governing equations considered are non-linear system of one- dimensional partial differential equations in Eulerian hydrodynamics. We analyze the system for possible analytical and numerical solutions. The analytical solution for the system is attempted using Lie group of point transformations. The arbitrary constants that appear in the expressions of generators of the Lie group transformations provide the possibility of cases with and without artificial viscosity. The symmetry variables which lead the governing system of non-linear partial differential equations to a reduced system of ordinary differential equations are determined using symmetry generators.  Numerical calculations have been performed to obtain the similarity exponent and the profiles of the flow variables using the forward difference and Lax-Friedrichs schemes. The effect of artificial viscosity on the flow variables, the process of stability and consistency along with the Courant- Friedrichs Lewy condition is investigated. The numerical solution by Lax-Friedrichs scheme found to be highly suitable for the system since the effect of flow variables, velocity and pressure in the presence of artificial viscosity parameter is negligible which agrees with the analytical solution obtained by Lie symmetry method.