The point explosion with heat conduction

Abstract

The influence of nonlinear heat conduction is investigated for strong point explosions in an ambient gas. An ideal gas equation of state and a heat conductivity depending on temperature and density in power-law form are assumed. It is shown that two spherical waves are obtained—a shock wave and a heat wave. They have sharp fronts which run at different speeds, in general, and in a relative order depending on parameters and time. Starting from the underlying Lie group symmetry, self-similar solutions of the problem are discussed in detail; they exist under the assumption that the ambient gas density decays with a given power of the radius. The non-self-similar situation, occurring for uniform density, is also considered. In this case, the shock front first runs behind the heat front, but then overtakes it at a certain time t1. For t≫t1, the well-known hydrodynamic solution of the problem without heat conduction becomes valid, except for a central, almost isobaric region where heat conduction modifies the classical result and keeps the temperature finite. It is shown that this central zone still has the form of a heat wave with a sharp front and evolves self-similarly, though with a smaller similarity exponent than the global hydrodynamic wave. Analytic results for the central temperature and the radial extension of the conduction dominated zone are given.