Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. II. Stability analysis

Conditions for the stability under linear perturbations around the homogeneous cooling state are studied for dilute granular gases of inelastic and rough hard disks or spheres with constant coefficients of normal ($\alpha$) and tangential ($\beta$) restitution. After a formally exact linear stability analysis of the Navier–Stokes–Fourier hydrodynamic equations in terms of the translational ($d_t$) and rotational ($d_r$) degrees of freedom, the transport coefficients derived in the companion paper [A. Megías and A. Santos, “Hydrodynamics of granular gases of inelastic and rough hard disks or spheres. I. Transport coefficients” Phys. Rev. E 104, 034901 (2021)] are employed. Known results for hard spheres [Garzó, Santos, and Kremer, Phys. Rev. E 97, 052901 (2018)] are recovered by setting $d_t=d_r=3$, while novel results for hard disks ($d_t=2, d_r=1$) are obtained. In the latter case, a high-inelasticity peculiar region in the $(\alpha ,\beta )$ parameter space is found, inside which the critical wave number associated with the longitudinal modes diverges. Comparison with event-driven molecular dynamics simulations for dilute systems of hard disks at $\alpha =0.2$ shows that this theoretical region of absolute instability may be an artifact of the extrapolation to high inelasticity of the approximations made in the derivation of the transport coefficients, although it signals a shrinking of the conditions for stability. In the case of moderate inelasticity ($\alpha =0.7$), however, a good agreement between the theoretical predictions and the simulation results is found.

Figure 1

Dispersion relations $\varpi \left(k\right)$ for the hydrodynamic modes vs the reduced wave number $k$. The curves correspond to the degenerate shear mode ${\varpi }_{\perp }$ (red lines), the heat mode ${\varpi }_{\parallel ,3}$, and the sound modes ${\varpi }_{\parallel ,1}$ and ${\varpi }_{\parallel ,2}$ (blue lines). Note that when ${\varpi }_{\parallel ,1}$ and ${\varpi }_{\parallel ,2}$ become a complex conjugate pair, only the (common) real part is plotted. The solid and dotted lines represent the HD system, while the dashed and dash-dotted lines refer to HS systems. The coefficient of normal restitution is $\alpha =0.7$, the reduced moment of inertia is $\kappa =\frac{1}{2}$ (HD) or $\kappa =\frac{2}{5}$ (HS), and the coefficients of tangential restitution are (a) $\beta =-0.5$, (b) $\beta =0$, (c) $\beta =0.5$, and (d) $\beta =1$.

Figure 2

Same as described in the caption of Fig. 1, except that $\alpha =0.2$.

Figure 3

(a) Plane $\alpha$ vs $\beta$ showing the locus ${\lambda }^{*}={\mu }^{*}$ for HD with a reduced moment of inertia $\kappa =0.1$, 0.3, 0.5, 0.8, and 1. In each case, $k_{\parallel }\to \infty$ in the region below the locus, which has an apex located at $(\alpha ,\beta )=({\alpha }_{\text{apex}},{\beta }_{\text{apex}})$. (b) Dependence of ${\alpha }_{\text{apex}}$ and ${\beta }_{\text{apex}}$ on $\kappa$ for HD and HS. (c) Variation with $\kappa$ of the area of the region where $k_{\parallel }\to \infty$ for HD and HS.

Figure 4

Density plots of the reduced critical wave number $k_{\perp }$ in the plane $\alpha$ vs $\beta$ for (a) HD with a uniform mass distribution ($\kappa =\frac{1}{2}$), (b) HD with a mass distribution concentrated on the outer surface ($\kappa =1$), (c) HS with a uniform mass distribution ($\kappa =\frac{2}{5}$), and (d) HS with a mass distribution concentrated on the outer surface ($\kappa =\frac{2}{3}$).

Figure 5

Same as described in the caption of Fig. 4, but for the reduced critical wave number $k_{\parallel }$.

Figure 6

Plane $\alpha$ vs $\beta$ showing the locus $k_{\perp }=k_{\parallel }$ for HD ($\kappa =\frac{1}{2}$ and 1) and HS ($\kappa =\frac{2}{5}$ and $\frac{2}{3}$). In each case, the longitudinal heat mode is the most unstable one ($k_{\parallel }>k_{\perp }$) in the region below the locus.

Figure 7

(a) Plot of the MD simulation values of the KLD $\mathcal{D}$ vs $\beta$ at $\alpha =0.2$ and $\alpha =0.7$ for system A (see Table 2). (b) Plot of the MD simulation values of the KLD $\mathcal{D}$ vs $\alpha$ at $\beta =0.25$ for the same system. (c) Theoretical eigenvalues ${\varpi }_{\perp }$ (transverse shear mode) and ${\varpi }_{\parallel ,3}$ (longitudinal heat mode) vs $\beta$ at $\alpha =0.2$ and $\alpha =0.7$ for a reduced wave number $k=1.253$. (d) Theoretical eigenvalues ${\varpi }_{\perp }$ (transverse shear mode) and ${\varpi }_{\parallel ,3}$ (longitudinal heat mode) vs $\alpha$ at $\beta =0.25$ for a reduced wave number $k=1.253$. The vertical dotted lines denote the borders of the regions where, according to theory, the HCS of the system is unstable (${\varpi }_{\parallel ,3}>0$) at (a, c) $\alpha =0.2$ and (b, d) $\beta =0.25$.

Figure 8

MD temporal evolution of (a) the rotational-to-translational temperature ratio $\theta$, (b) the excess translational velocity kurtosis $a_{20}$, (c) the excess angular velocity kurtosis $a_{02}$, (d) the translational-angular correlation cumulant $a_{11}$, (e) the KLD $\mathcal{D}$, and (f) the ratio $4\mathcal{D}/{({\sigma }_{\text{cell}}^2/a_{\text{cell}}-1)}^2$ [see Eq. (27)]. The (blue) thin and (red) thick lines correspond to systems A and B, respectively (see Table 2), in both cases with $\alpha =0.2$ and $\beta =0.25$. The horizontal dashed lines in panels (a–d) are theoretical values [24].

Figure 9

MD snapshots at $s=100$ collisions per particle showing the positions [left panels (a, c)] and translational velocities [right panels (b, d)] in the cases of systems A [top panels (a, b)] and B [bottom panels (c, d)] (see Table 2), in both cases with $\alpha =0.2$ and $\beta =0.25$. In the left panels, the color code refers to the ratio between the rotational and the translational kinetic energies of each particle.

Figure 10

MD temporal evolution of (a) the rotational-to-translational temperature ratio $\theta$, (b) the excess translational velocity kurtosis $a_{20}$, (c) the excess angular velocity kurtosis $a_{02}$, (d) the translational-angular correlation cumulant $a_{11}$, (e) the KLD $\mathcal{D}$, and (f) the ratio $4\mathcal{D}/{({\sigma }_{\text{cell}}^2/a_{\text{cell}}-1)}^2$ [see Eq. (27)]. The (blue) thin solid, the (red) thin dashed, and the (green) thick dash-dotted lines correspond to systems A, B, and C, respectively (see Table 2), in the three cases with $\alpha =0.7$ and $\beta =0.25$. The horizontal dashed lines in panels (a–d) are theoretical values [24].

Figure 11

MD snapshots at $s=100$ collisions per particle showing the positions [left panels (a, c, e)] and translational velocities [right panels (b, d, f)] in the cases of systems A [top panels (a, b)], B [middle panels (c, d)], and C [bottom panels (e, f)] (see Table 2), in the three cases with $\alpha =0.7$ and $\beta =0.25$. In the left panels, the color code refers to the ratio between the rotational and the translational kinetic energies of each particle.