Blast waves in the zero temperature hard sphere gas: double scaling structure

Abstract

We study the blast generated by sudden localized release of energy in a cold gas. Specifically, we consider one-dimensional hard-rod gas and two-dimensional hard disc gas. For this problem, the Taylor-von Neumann-Sedov (TvNS) solution of Euler equations has a self-similar form. The shock wave remains infinitely strong for the zero-temperature gas, so the solution applies indefinitely. The TvNS solution ignores dissipation, however. We show that this is erroneous in the core region which, in two dimensions, expands as $t^{2/5}$ while the shock wave propagates as $t^{1/2}$. A new self-similar solution depending on the scaling variable $r/t^{2/5}$ describes the core, while the TvNS solution describes the bulk. We demonstrate this from a numerical solution of the Navier-Stokes (NS) equations and from molecular dynamics simulations for a gas of hard discs in two dimensions and hard rods in one dimension. In both cases, the shock front position predicted by NS equations and by the TvNS solution agrees with that predicted by molecular dynamics simulations. However, the NS equations fail to describe the near-core form of the scaling functions.