Attempted density blowup in a freely cooling dilute granular gas: Hydrodynamics versus molecular dynamics

It has been recently shown [I. Fouxon et al., Phys. Rev. E 75, 050301(R) (2007); I. Fouxon et al., Phys. Fluids 19, 093303 (2007)] that, in the framework of ideal granular hydrodynamics (IGHD), an initially smooth hydrodynamic flow of a granular gas can produce an infinite gas density in a finite time. Exact solutions that exhibit this property have been derived. Close to the singularity, the granular gas pressure is finite and almost constant. We report molecular dynamics (MD) simulations of a freely cooling gas of nearly elastically colliding hard disks, aimed at identifying the “attempted” density blowup regime. The initial conditions of the simulated flow mimic those of one particular solution of the IGHD equations that exhibits the density blowup. We measure the hydrodynamic fields in the MD simulations and compare them with predictions from the ideal theory. We find a remarkable quantitative agreement between the two over an extended time interval, proving the existence of the attempted blowup regime. As the attempted singularity is approached, the hydrodynamic fields, as observed in the MD simulations, deviate from the predictions of the ideal solution. To investigate the mechanism of breakdown of the ideal theory near the singularity, we extend the hydrodynamic theory by accounting separately for the gradient-dependent transport and for finite density corrections.

Figure 1

The initial values of the hydrodynamic fields in the example of attempted density blowup considered in this work. Shown are the rescaled initial density and velocity of the gas [see Eqs. (15, 17)] versus the rescaled Eulerian coordinate $x/l$ along the channel. Only the right half of the system is shown. The values of ${\rho }_0$, $T_0$, and $l$, used in our molecular dynamic simulations, are presented in the beginning of Sec. 3D.

Figure 2

(Color online) Evolution of the density $\rho \left(m,t\right)$ in the Lagrangian frame at initial times. Circles: the results of MD simulations. Solid line: the prediction of the analytic solution, first equation of Eqs. (21), back to the physical variables. Dashed line: a numerical solution of the nonideal hydrodynamic equations (8, 9, 10), which account for the gradient-dependent transport. The dashed line is indistinguishable from the solid line.

Figure 3

(Color online) Evolution of the density $\rho \left(m,t\right)$ in the Lagrangian frame at late times. The $m$ axis is zoomed into in order to focus on the high-density region. Circles: the results of MD simulations. Solid line: the prediction of the first equation of Eqs. (21), back to the physical variables. Dashed line: a numerical solution of the nonideal hydrodynamic equations (8, 9, 10), which account for the gradient-dependent transport. The dash-dotted line marks the close-packing density ${\rho }_c=2/\left(\sqrt{3}{\sigma }^2\right)$.

Figure 4

(Color online) Evolution of the longitudinal velocity $v\left(m,t\right)$ in the Lagrangian frame. Circles: the results of MD simulations. Solid line: the prediction of Eq. (22), back to the physical variables. Dashed line: a numerical solution of the nonideal hydrodynamic equations (8, 9, 10), which account for the gradient-dependent transport. The dashed line is indistinguishable from the solid line.

Figure 5

(Color online) Evolution of the (minus) longitudinal velocity $v\left(m,t\right)$ in the Lagrangian frame, in a semilogarithmic scale. Circles: the results of MD simulations. Solid line: the prediction of Eq. (22), back to the physical variables. The dashed line: a numerical solution of the nonideal hydrodynamic equations (8, 9, 10), which account for the gradient-dependent transport.

Figure 6

(Color online) Evolution of the gas pressure, obtained using the ideal equation of state, $p\left(m,t\right)=\rho \left(m,t\right)T\left(m,t\right)$, in the Lagrangian frame. Circles: the results of MD simulations. Solid line: the prediction of the second equation of Eqs. (21), back to the physical variables. Dashed line: a numerical solution of the nonideal hydrodynamic equations (8, 9, 10), which account for the gradient-dependent transport.

Figure 7

(Color online) The time history of the density maximum $\rho \left(0,t\right)$ and the temperature minimum $T\left(0,t\right)$. The symbols show the results of MD simulations: black circles depict the quantity $1-\sqrt{{\rho }_0/\rho \left(0,t\right)}$, and green diamonds depict the quantity $1-\sqrt{T\left(0,t\right)}$. The solid line is the prediction of Eq. (35): an immediate consequence of Eq. (24). The dashed line is a numerical solution of the nonideal hydrodynamic equations (8, 10), which account for the gradient-dependent transport. The inset zooms in at later times.

Figure 8

(Color online) Decay of the total energy per particle, $E\left(t\right)/N^{\prime }$, as measured in the MD simulations (circles) and predicted by the exact solution (solid line). The time derivative ${\left(N^{\prime }\right)}^{-1}dE/dt$, displayed in the inset (the notation for the circles and line is the same), shows that the energy decay remains smooth at all simulated times.

Figure 9

(Color online) The density profiles for $t/\tau =0.944$ (a) and $t/\tau =0.966$ (b) as obtained from the MD simulations (circles), the analytic solution, Eq. (21), of the IGHD equations (dash-dotted line), the numerical solution of the dilute NIGHD equations (dashed line), and the numerical solution of finite-density hydrodynamic equations with the gradient-dependent transport terms neglected (solid line).