Assessing a hydrodynamic description for instabilities in highly dissipative, freely cooling granular gases

An intriguing phenomenon displayed by granular flows and predicted by kinetic-theory-based models is the instability known as particle “clustering,” which refers to the tendency of dissipative grains to form transient, loose regions of relatively high concentration. In this work, we assess a modified-Sonine approximation recently proposed [Garzó, Santos, and Montanero, Physica A 376, 94 (2007)] for a granular gas via an examination of system stability. In particular, we determine the critical length scale associated with the onset of two types of instabilities—vortices and clusters—via stability analyses of the Navier-Stokes-order hydrodynamic equations by using the expressions of the transport coefficients obtained from both the standard and the modified-Sonine approximations. We examine the impact of both Sonine approximations over a range of solids fraction $\phi <0.2$ for small restitution coefficients $e$ $=$ 0.25–0.4, where the standard and modified theories exhibit discrepancies. The theoretical predictions for the critical length scales are compared to molecular dynamics (MD) simulations, of which a small percentage were not considered due to inelastic collapse. Results show excellent quantitative agreement between MD and the modified-Sonine theory, while the standard theory loses accuracy for this highly dissipative parameter space. The modified theory also remedies a high-dissipation qualitative mismatch between the standard theory and MD for the instability that forms more readily. Furthermore, the evolution of cluster size is briefly examined via MD, indicating that domain-size clusters may remain stable or halve in size, depending on system parameters.

Figure 1

Spatially averaged velocity fields from MD simulations with $e$ $=$ 0.4, $\phi$ $=$ 0.05, and number of particles $N$ $=$ 2096 at (a) five collisions per particle (cpp) and (b) 90 cpp, and (c) corresponding Fourier momentum spectra ($P$) normalized to the value at $k$ $=$ 4$\pi$. Square cells with length $L$/5 were averaged for visualization in (a) and (b).

Figure 2

Snapshots of particle positions (showing only a slice of thickness $L/10$) extracted from three-dimensional MD simulations with $e$ $=$ 0.6, $\phi$ $=$ 0.2, $N$ $=$ 2000 at (a) 2 cpp, (b) 40 cpp, (c) 800 cpp, and (d) corresponding Fourier mass spectra ($R$).

Figure 3

Critical clustering length scale ($L_{\mathrm{Cluster}}/d$) as a function of solids fraction for (a) $e$ $=$ 0.25 and (b) $e$ $=$ 0.4. The solid and dashed lines correspond to the modified and standard theories, respectively. The data points correspond to MD.

Figure 4

Critical length scale for vortex and cluster instabilities (i.e., $L_{\mathrm{Vortex}}/d$ and $L_{\mathrm{Cluster}}/d$, respectively) plotted as a function of solids fraction for $e$ $=$ 0.25. The solid and dashed lines correspond to the modified and standard theories, respectively.