An Exact Solution for the Propagation of Shock Waves in Self-Gravitating Perfect Gas in the Presence of Magnetic Field and Radiative Heat Flux

Abstract

Propagation of spherical shock wave with azimuthal magnetic field and radiation heat flux in self-gravitating perfect gas is investigated. The azimuthal magnetic field and the initial density are assumed to vary according to power law. An exact similarity solution is reported when loss of energy due to radiation escape is notable and radiation pressure is non-zero. The entire energy of the shock wave is varying and increases with time. The effects of variation of the radiation pressure number, the initial density variation index, the Alfven-Mach number, the gravitational parameter and the adiabatic exponent are workout in detail. The shock strength increases with an increase in the initial density variation index. On the other hand, presence of magnetic field or an increment in the value of the radiation pressure number or the ratio of specific heats or gravitational parameter the shock strength decreases. It is obtained that increase in the radiation pressure number and gravitational parameter has same behavior on the flow variables. Also, it is observed that an increase in the value of gravitational parameter and the adiabatic exponent have same behavior on the fluid velocity, the material pressure, the radiation pressure, the mass and the radiation flux and azimuthal magnetic field.

An exact solution for the propagation of cylindrical shock waves in a rotational axisymmetric non-ideal gas with axial magnetic field and radiative heat flux

“…We have calculated the values of flow variables from equations (29), (30), (36) -(38) and the results are shown in figures 1, 2 and Table 1. The values of the physical parameters for calculations are taken to be = 4/3,5/3; −2 = 0.3,0.35; and = 0,0.05,0.1; ( [14,15,20,[26][27][28][29][30][31][32][33]). For fully ionized gas = 5/3 and for relativistic gases = 4/3.…”

Exact Self-similar Solution for Spherical Shock Wave in a Non-ideal Gas with Azimuthal Magnetic Field under Isothermal Flow Condition

“…where, is the gas constant, is the ratio of specific heats, is the internal energy per unit mass of the gas, = , on the density of the gas. For the self-similar solution [6], the shock velocity = is assumed to vary as [15][16][17]:…”

“…To obtain the self-similar solution, the unknown variables are written in the following form [13,15] = ( ), = ( ), = 2 ( ),…”

“…The radiation heat transfer equations are not used explicitly, whereas the radiation flux and other flow variables in the flow field behind the shock are evaluated from the equation of motion by the use of "product solution" of Mc Vittie [12]. Other important work related to shock wave problem using the "product solution" of Mc Vittie et al [12][13][14][15][16][17][18]. Nath et al [17] have studied the shock wave propagation in a self-gravitating perfect gas with variable density in the presence of magnetic field in isothermal flow condition and they obtained the exact similarity solutions using the similarity method [12].…”

“…In all of the works mentioned above and known to us, the exact self-similar solution have not investigated by any of the authors using the "product solution" of Mc Vittie [12] for the propagation of shock wave in a non-ideal gas with azimuthal magnetic field in the case of isothermal flow. The motivation for the present study is the pioneering work on exact solution on shock wave propagation by Mc Vittie et al [12,13,[15][16][17][18]. In the present study we adopted van der Waal gas model to obtain an exact self-similar solution for the unsteady isothermal flow behind a spherical shock wave in a non-ideal gas with azimuthal magnetic field with constant initial density.…”