An Approximate Analytical Solution of Imploding Strong Shocks in a Non-Ideal Gas through Lie Group Analysis

Abstract

The method of Lie group transformation is used to obtain an approximate analytical solution to the system of first-order quasilinear partial differential equations that govern a one-dimensional unsteady planer, cylindrically symmetric and spherically symmetric motion in a non-ideal gas, involving strong shock waves. Invariance groups admitted by the governing system of partial differential equations, which are indeed continuous group of transformations under which the system of partial differential equations remains invariant, are determined, and the complete Lie algebra of infinitesimal symmetries is established. The infinitesimal generators are used to construct the similarity variables. These similarity variables are used to reduce the governing system of partial differential equations into a system of ordinary differential equations.

Similarity Solutions for Strong Shocks in a Non-Ideal Gas

“…A theoretical study of the imploding shock wave near the center of convergence, in an ideal gas was first performed by Guderley [7]. Among the extensive work that followed, we mention the contributions of Sakurai [17], Zeldovich and Raizer [23], Hayes [9], Ames [1], Axford and Holm [2,3], Lazarus [11], Hafner [8], Sharma and Radha [19], Jena and Sharma [10], Conforto [6], Madhumita and Sharma [14], Sharma and Arora [18], Sharma and Radha [20] and Singh et al [13] who presented high accuracy results and alternative approaches for the investigation of implosion problem. Steeb [21] determined the similarity solutions of the Euler equations and the Navier-Stokes equations for incompressible flows using the group theoretic approach outlined in the work of Bluman and Cole [4], Ovasiannikov [16], Olver [15], Logan [12] and Bluman and Kumei [5].…”

Analytical Solution of the Blast Wave Problem in a Non-Ideal Gas

“…Sachdev et al [3] generalized the work of McVitte [2] and presented the exact solutions of compressible gasdynamic equations with spherical geometry. Singh et al [4] used the method of Lie group transformations to obtain an approximate analytical solution to the system of the first-order quasi-linear partial differential equations that govern a one-dimensional unsteady planar and non-planar motion in a non-ideal gas, involving strong shock waves. Oliveri and Speciale [5] used the substitution principle to obtain an exact solution for the unsteady equation of perfect gas.…”

Magnetogasdynamic shock waves in a nonideal self-gravitating gas using Lie group theoretic method

“…Bluman and Kumei (1989) and Olver (1986) have discussed the theoretical portion of the Lie group of transformation and its applications in numerous domains for solving problems. By using the Lie group theoretic method, Singh et al . (2010) have studied the propagation of strong shock wave into a nonideal gas and derived the approximate analytical solution.…”