Self-organized shocks in the sedimentation of a granular gas

A granular gas in gravity heated from below develops a certain stationary density profile. When the heating is switched off, the granular gas collapses. We investigate the process of sedimentation using computational hydrodynamics, based on the Jenkins-Richman theory, and find that the process is significantly more complex than generally acknowledged. In particular, during its evolution, the system passes several stages which reveal distinct spatial regions of inertial (supersonic) and diffusive (subsonic) dynamics. During the supersonic stages, characterized by $\text{Mach}>1$, the system develops supersonic shocks which are followed by a steep front of the hydrodynamic fields of temperature and density, traveling upward.

Figure 1

Stationary profiles of packing fraction, $\varphi \left(y\right)$, and temperature, $T\left(y\right)$, of a granular system heated from below, $T_0={10}^6$, establishing the initial conditions of the sedimentation. Height is given in units of $\sigma$.

Figure 2

Total energy as a function of time for different values of the initial bottom plate temperature, $T_0$. Dashed lines indicate the stages of the sedimentation process.

Figure 3

Evolution of the hydrodynamic fields. Red circles indicate the vertical location where $\text{Mach=1}$. Scale of the vertical axis is the same for all graphs (here given only for $\text{Mach}$). Height is given in units of $\sigma$.

Figure 4

Location of the maximal slope of the temperature and density profiles, as a function of time. Circles indicate $\text{Mach=1}$. Height is given in units of $\sigma$.

Figure 5

Decay of the total energy for different coefficient of restitution ($T_0={10}^7$).