Evolution and stability of shock waves in dissipative gases characterized by activated inelastic collisions

Abstract

Previous experiments have revealed that shock waves driven through dissipative gases may become unstable, for example, in granular gases, and in molecular gases undergoing strong relaxation effects. The mechanisms controlling these instabilities are not well understood. We successfully isolated and investigated this instability in the canonical problem of piston driven shock waves propagating into a medium characterized by inelastic collision processes. We treat the standard model of granular gases, where particle collisions are taken as inelastic with constant coefficient of restitution. The inelasticity is activated for sufficiently strong collisions. Molecular dynamic simulations were performed for 30,000 particles. We find that all shock waves investigated become unstable, with density non-uniformities forming in the relaxation region. The wavelength of these fingers is found comparable to the characteristic relaxation thickness. Shock Hugoniot curves for both elastic and inelastic collisions were obtained analytically and numerically. Analysis of these curves indicate that the instability is not of the Bethe-Zeldovich-Thompson or Dyakov-Kontorovich types. Analysis of the shock relaxation rates and rates for clustering in a convected fluid element with the same thermodynamic history outruled the clustering instability of a homogeneous granular gas. Instead, wave reconstruction of the early transient evolution indicates that the onset of instability occurs during the re-pressurization of the gas following the initial relaxation of the medium behind the lead shock. This re-pressurization gives rise to internal pressure waves in the presence of strong density gradients. This indicates that the mechanism of instability is more likely of the vorticity-generating Richtmyer-Meshkov type, relying on the action of the inner pressure waves development during the transient relaxation.