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.

中文翻译：

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非弹性和粗糙的硬盘或球体的颗粒气体的流体动力学。二、稳定性分析

研究了非弹性和粗糙硬盘或球体的稀颗粒气体在均匀冷却状态周围线性扰动下的稳定性条件（$\alpha$) 和切线 ($\beta$) 恢复原状。在对 Navier-Stokes-Fourier 流体动力学方程进行形式化精确线性稳定性分析后$d_{吨}$) 和旋转 ($d_r$) 自由度，在配套论文 [A. Megías 和 A. Santos，“非弹性和粗糙硬盘或球体的颗粒气体的流体动力学。I. 传输系数“ Phys. Rev. E 104 , 034901 (2021)]。硬球的已知结果 [Garzó、Santos 和 Kremer，Phys. Rev. E 97 , 052901 (2018)] 通过设置恢复$d_{吨}=d_r=3$, 而硬盘的新结果 ($d_{吨}=2, d_r=1$) 获得。在后一种情况下，一个高非弹性的特殊区域$(\alpha ,\beta )$找到参数空间，其中与纵模相关的临界波数发散。与事件驱动的分子动力学模拟对硬盘稀释系统的比较$\alpha =0.2$表明这个绝对不稳定性的理论区域可能是外推到推导传输系数的近似值的高非弹性的产物，尽管它表明稳定性条件的缩小。在中等无弹性的情况下（$\alpha =0.7$)，然而，发现理论预测和模拟结果之间有很好的一致性。