PROPAGATION OF NONLINEAR SHOCK WAVES IN A NON-IDEAL DUSTY GAS WITH THERMAL RADIATION UNDER THE OPTICALLY THIN APPROXIMATION

Abstract

A smooth, finite amplitude disturbance propagating and ultimately becoming a shock in a one-dimensional non-ideal radiative gas is considered. A single generalized-geometry parameter is used for planer, cylindrical and spherical. The gas is considered to be a simplified van der Waals gas, in which molecular co-volume is retained and intermolecular attraction is ignored. Thermal radiation is treated as a local source which is proportional to the difference between the fourth powers of the gas and the ambient temperature, optically thin. The basic equations for the conservation of mass, momentum, and pressure are represented as a quasilinear hyperbolic system with sources. There are three characteristic speeds: the material speed, u-a, and u+a, the two acoustic speeds. A formula based on the principle of wavefront expansion is then employed to understand the evolution of the amplitude of the leading front and when the smooth solution loses regularity, and to understand how the wavefront steepens with an initially smooth compressive disturbance. The analysis defines separate roles of the governing parameters. The non-ideal sound speed introduces the molecular co-volume to the acoustic matrix, while source terms resulting from radiation or geometry, modify the state transported along the characteristics. For admissible states, the local sound speed is increasing with the co-volume at fixed pressure and density. The radiative term provides cooling for states warmer than the equilibrium state of the atmosphere, and warming for cooler states, and the strength of the cooling or warming depends on a dimensionless radiation parameter. The cylindrical and spherical geometry have increasingly stronger radial source terms than planar flow. Together, they control the amplitude growth and decay as well as the critical propagation distance to shock formation. The formulation allows for evidence-informed future computations.