Shock waves in dense hard disk fluids

Abstract

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.

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

“…Figure instantaneous structure, the peak temperature and temperature at the piston are tracked. Initially the temperature jump of the shock is approximately u2 400, as predicted for a system of elastic disks [33]. The temperature measured at the piston sur face decays until coming to a quasiequilibrium state, at which point most inelastic collisions have subsided; note that the kinetic model taken maintains an exponentially small fraction of activated collisions as the temperature decays below the acti vation temperature.…”

“…The initial packing factor of the disks was chosen to be ri ( = (Nnd2)/4A = 0.012, where N nd2/4 is the volume (area) of the N hard disk with diameter d, and A = Lx x L v the domain area; the initial gas is thus in the ideal-gas regime [33]. All distances have been normalized by the initial mean free path of the system of disks X|, which takes the form [30] y /2 T c d n xb 2{ r i\ ) '…”

“…They represent the trajectories of fluid particles, right running pressure waves, and left running pressure waves, respectively [15]. The scaled speed of sound for such a medium is approximated for an elastic system of disks, taken as [33] -…”

“…This confirms that up/u* is a scaling parameter for the dynamics. In our parametric study, we henceforth maintain u* -10 and vary only up and s. We also set the initial packing factor t]\ = 0 .1 2 , a parameter we do not explore in the present study; see Sirmas et al for its effect on the shock jump conditions in the case of nondissipative collisions [33],…”

Evolution of shock instability in granular gases with viscoelastic collisions

“…The scaled speed of sound for such a media is approximated for an elastic system of disks, taken as [16]:…”