Propagation of a spherical shock wave in mixture of non-ideal gas and small solid particles under the influence of gravitational field with conductive and radiative heat fluxes

Abstract

Self-similar solutions are obtained for one-dimensional unsteady adiabatic flow behind a spherical shock wave propagating in a dusty gas with conductive and radiative heat fluxes under the influence of a gravitational field. The shock is assumed to be driven out by a moving piston and the dusty gas to be a mixture of non-ideal gas and small solid particles, in which solid particles are uniformly distributed. It is assumed that the equilibrium flow-conditions are maintained and variable energy input is continuously supplied by the piston. The heat conduction is expressed in terms of Fourier’s law and the radiation is considered to be of the diffusion type for an optically thick grey gas model. The thermal conductivity $K$ and the absorption coefficient $\alpha_{R}$ are assumed to vary with temperature and density. The medium is assumed to be under the influence of a gravitational field due to central mass $( \bar{m} )$ at the origin (Roche Model). It is assumed that the gravitational effect of the mixture itself can be neglected compared with the attraction of the central mass. The initial density of the ambient medium is taken to be always constant. The effects of the variation of the gravitational parameter and nonidealness of the gas in the mixture are investigated. Also, the effects of an increase in (i) the mass concentration of solid particles in the mixture and (ii) the ratio of the density of solid particles to the initial density of the gas on the flow variables are investigated. It is shown that due to an increase in the gravitational parameter the compressibility of the medium at any point in the flow-field behind the shock decreases and all other flow variables and the shock strength are increased. Further, it is found that the presence of gravitational field increases the compressibility of the medium, due to which it is compressed and therefore the distance between the piston and the shock surface is reduced. The shock waves in dusty gas under the influence of a gravitational field can be important for description of shocks in supernova explosions, in the study of central part of star burst galaxies, nuclear explosion, star formation in shocks and shocks in stellar explosion, rupture of a pressurized vessels and explosion in the ionosphere etc. Also, the solution obtained can be used to interpret measurements carried out by spacecraft in the solar wind and in neighborhood of the Earth’s surface.