ABSTRACT
An explosively generated shock wave with time-dependent radius R(t) is characterized by a phase in which the shocked gas becomes radiative with an effective adiabatic index γ ≃ 1. Using the result that the post-shock gas is compressed into a shell of width ΔR/R ≃ δ, where δ = γ − 1, we show that a choice of self-similar variable that exploits this compressive behaviour in the limit that γ → 1 naturally leads to a series expansion of the post-shock fluid density, pressure, and velocity in the small quantity δ. We demonstrate that the leading-order (in δ) solutions, which are increasingly accurate as γ → 1, can be written in simple, closed forms when the fluid is still approximated to be in the energy-conserving regime (i.e. the Sedov–Taylor limit), and that the density declines exponentially rapidly with distance behind the shock. We also analyse the solutions for the bubble surrounding a stellar or galactic wind that interacts with its surroundings, and derive expressions for the location of the contact discontinuity that separates the shocked ambient gas from the shocked wind. We discuss the implications of our findings in the context of the dynamical stability of nearly isothermal shocks.
The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas
