Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field

Abstract

The topic "converging shock waves" is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on shock waves should be taken in account. We have considered a problem of propagation of cylindrical shock waves in real medium. Magnetic field has been assumed in axial direction. It has been assumed that electrical resistance is zero. The problem can be represented by a system of hyperbolic Partial Differential Equations (PDEs) with jump conditions at the shock as the boundary conditions. The Lie group theoretic method has been used to find solutions to the problem. Lie's symmetric method is quite useful as it reduces one-dimensional flow represented by a system of hyperbolic PDEs to a system of Ordinary Differential Equations (ODEs) by means of a similarity variable. Infinitesimal generators of Lie's group transformation have been obtained by invariant conditions of the governing and boundary conditions. These generators involves arbitrary constants that give rise to different possible cases. One of the cases has been discussed in detail by writing reduced system of ODEs in matrix form. Cramer's rule has been used to find the solution of system in matrix form. The results are presented in terms of figures for different values of parameters. The effect of non-ideal medium on the flow has been studied. Guderley's rule is used to compute similarity exponents for cylindrical shock waves, in gasdynamics and in magnetogasdynamics (ideal medium), in order to set up a comparison with the published work. The computed values are very close to the values in published articles.Raizer [3] have given simple scaling arguments to obtain similarity solutions with an illustration of self-similar or invariant nature of the scaled solutions. Furthermore, the great application of their work is given by Barenblatt [4], giving a clear explanation of the nature of invariant solutions of the first and second kind.A large number of natural phenomena [5][6][7], magnetized stellar winds, shapes of planetary nebulae, galactic winds, complex filamentary structures in molecular clouds, synchrotron radiation from supernova remnants, gamma-ray bursts, dynamo effects in stars, galaxies and galaxy clusters, structure and interaction of supershells as well as other interesting problems all have important roles in magnetic field. Due to its theoretical and practical importance in the variety of astrophysical situations, plasma physics, nuclear science and engineering physics, analysis of magnetogasdynamics has received more attention from researchers in the fields ranging from condensed matter to gas dynamics. Lie's approach is applied to estimate the structure of strong imploding shock waves. Guderley [8], Zeldovich and Raizer [3], Sedov [2], Korobeinikov [9] and Arora et al. [10], etc. studied the imploding shock waves and contributed remarkably in this field. Steeb [11] has applied a group...

Plane one-dimensional MHD flows: Symmetries and conservation laws

“…The research concerning with MHD in cylindrical geometry is devoted to exact and approximate solutions as well as to efficient computational approaches for solving practical problems [6,7,8,9,10,11]. Particular attention is devoted to the analysis of cylindrical shock waves [12,13,14,15,16,17,18,19,20,21,19,22,23].…”

Similarity solutions for cylindrical shock waves in a non-ideal gas under the action of monochromatic radiation

“…The basic concept of the Lie group of transformations can be found in [25,26]. Latest literature related to the application of Lie group method can be found in the works of Logan and Perez [27], Hydon [28], Singh and Arora [29], Devi et al [30], Singh et al [31], Sharma and Arora [32] and Jena [33], etc.…”