Media composed of colliding hard disks (2D) or hard spheres (3D) serve as good approximations for the collective hydrodynamic description of gases, liquids and granular media. In the present study, the compressible hydrodynamics and shock dynamics are studied for a two-dimensional hard-disk medium at both the continuum and discrete particle level descriptions. For the continuum description, closed form analytical expressions for the inviscid hydrodynamic description, shock Hugoniot, isentropic exponent and shock jump conditions were obtained using the Helfand equation of state. The closed-form analytical solutions permitted us to gain physical insight into the role of the material's density on its compressibility, i.e. how the medium compresses under mechanical loadings and sustains wave motion. Furthermore, the predictions were found in excellent agreement with calculations using the event driven molecular dynamics method involving 30,000 particles over the entire range of compressibility spanning the dilute ideal gas and liquid phases. In all cases, it was found that the energy imparted by the piston motion to the thermalized medium behind the propagating shock was quasi-independent of the medium's packing fraction, with a correction vanishing with increasing shock Mach numbers.

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 a 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 nonuniformities forming in the relaxation region. The wavelength of these fingers is found to be comparable to the characteristic relaxation thickness. Shock Hugoniot curves for both elastic and inelastic collisions were obtained analytically and numerically. Analysis of these curves indicates that the instability is not of the Bethe-Zeldovich-Thompson or D'yakov-Kontorovich type. Analysis of the shock relaxation rates and rates for clustering in a convected fluid element with the same thermodynamic history ruled out the clustering instability of a homogeneous granular gas. Instead, wave reconstruction of the early transient evolution indicates that the onset of instability occurs during repressurization of the gas following the initial relaxation of the medium behind the lead shock. This repressurization 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 wave development during the transient relaxation.

Previous experiments have revealed that shock waves driven through dissipative media may become unstable, for example, in granular gases, and in molecular gases undergoing strong relaxation effects. The current paper addresses this problem of shock stability at the Euler and Navier–Stokes continuum levels in a system of disks (two-dimensional) undergoing activated inelastic collisions. The dynamics of shock formation and stability is found to be in very good agreement with earlier molecular dynamic simulations (Sirmas & Radulescu, Phys. Rev. E, vol. 91, 2015, 023003). It was found that the modelling of shock instability requires the introduction of molecular noise for its development and sustenance. This is confirmed in two stability problems. In the first, the evolution of shock formation dynamics is monitored without noise, with only initial noise and with continuous molecular noise. Only the latter reproduces the results of shock instability of molecular dynamics simulations. In the second problem, the steady travelling wave solution is obtained for the shock structure in the inviscid and viscous limits and its nonlinear stability is studied with and without molecular fluctuations, again showing that instability can be sustained only in the presence of fluctuations. The continuum results show that instability takes the form of a rippled front of a wavelength comparable with the relaxation thickness of the steady shock wave, at scales at which molecular fluctuations become important, in excellent agreement with the molecular dynamic simulations.

The problem of thermal ignition in a homogeneous gas is revisited from a molecular dynamics perspective. A two-dimensional model is adopted, which assumes reactive disks of type A and B in a fixed domain that react to form type C products if an activation threshold for impact is surpassed. Such a reaction liberates kinetic energy to the product particles, representative of the heat release. The results for the ignition delay are compared with those obtained from the continuum description with the reaction rate evaluated from kinetic theory assuming local thermodynamic equilibrium and Maxwell-Boltzmann statistics, in order to assess the role played by molecular fluctuations. Ignition times obtained using molecular dynamics are ensemble averaged over 100 simulations to address the statistics of the ignition event. Results show two regimes of non-equilibrium ignition whereby ignition occurs at different times as compared to that for homogeneous ignition assuming local equilibrium. The first regime is at low activation energies, where the ignition time is found to be higher than that expected from theory for all values of heat release. The lower reaction rate is shown to occur due to a departure from local equilibrium for the different species, in agreement with predictions from Prigogine and Xhrouet. In this low activation energy regime, the ignition times from molecular dynamics are also found to be independent of domain size and there is little variance between different realizations under similar conditions, which suggests that the ignition is spa-Email addresses: nsirmas@uottawa.ca (N. Sirmas), matei@uottawa.ca (M. I. Radulescu)

Shocks in granular media have been shown to develop instabilities. We address the role that early stages of shock development have on this type of instability. We look at the evolution of shock waves driven by a piston in a dilute system of smooth inelastic disks, using both discrete particle and continuum modelling. To mimic a realistic granular gas, viscoelastic collisions are approximated with an impact velocity threshold u * needed for inelastic collisions to occur. We show that behaviour of the shock evolution is dependent on the ratio of piston velocity to impact velocity threshold u p /u * , and the coefficient of restitution ε. For u p /u * = 2.0, we recover shock evolution behaving similar to that observed in purely inelastic media. This is characterized by a short period where the shock front pulls towards the piston before attaining a developed structure. No pullback is seen for u p /u * = 1.0. Results show the onset of instability for these stronger shocks during this evolving stage. These results suggest that the early stages of shock evolution play an important role in the shock instability.