This paper uses a fluid‐mechanical model of a granular medium to calculate the hydrodynamic modes of a spatially uniform basic state. These modes are the granular analogs of the heat, sound, and shear modes of the standard fluid. Attention is focused on the possibility of an unstable mode that might result in the spontaneous development of inhomogeneities in density. Two cases are considered: the cooling medium which loses energy without replenishment, and the heated medium which reaches a steady state when an energy source balances the loss of energy through particle collisions. The spatially uniform state of the cooling granular medium is unstable. Two modes, analogous to the shear and heat conduction modes of a standard fluid, are unstable at long wavelengths. The growth of these modes is algebraic, rather than exponential, in time. The shear mode does not involve the formation of density inhomogeneities, but the heat mode does. At long wavelengths the heat mode can be visualized by imagining a converging velocity increasing the density of particles in a certain region. The increased collisional dissipation of granular thermal energy reduces the pressure, and prevents it from reversing the convergent velocities, so the condensation is not checked. The stability of the heated granular medium depends on the energy source. If the energy source selectively deposits energy in hot regions of a disturbance, the diffusion and collisional damping can be overwhelmed, and the disturbance grows exponentially. The standard fluid (completely elastic particles) can be recovered as a special case of the heated granular medium. In all cases, waves analogous to the sound and heat conduction modes are present. In some cases, a second type of sound wave is present at long wavelengths with the peculiar property of being damped more quickly for more elastic particles.

中文翻译：

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均匀颗粒介质的流体动力模式

本文使用流体力学 模型 颗粒介质来计算 流体动力 一个模式 在空间上统一的基本状态。这些模式是标准流体的热量，声音和剪切模式的颗粒类似物。注意集中在不稳定模式的可能性上，该模式可能导致密度不均匀性的自然发展。考虑两种情况：冷却介质损失能源 无需补给，并且加热的介质在 能源 源平衡损失 能源通过粒子碰撞。的在空间上冷却颗粒状介质的均匀状态是不稳定的。两种模式，类似于剪切和热传导标准流体的模式在长波长下不稳定。这些模式的增长在时间上是代数的，而不是指数的。剪切模式不涉及密度不均匀性的形成，但是加热模式涉及。在长波长下，可以通过想象会聚速度来提高特定区域中颗粒密度的方式来观察加热模式。颗粒热的碰撞耗散增加能源 降低压力并防止其逆转收敛速度，因此 缩合未检查。加热的颗粒介质的稳定性取决于能源资源。如果能源 源选择性沉积 能源 在热闹地区， 扩散碰撞阻尼会不堪重负，扰动呈指数增长。标准液（完全有弹性颗粒）可以作为加热的粒状介质的特殊情况进行回收。在所有情况下，波浪 类似于声音和 热传导模式存在。在某些情况下，第二类声波会以很长的波长存在，属性 更快地被阻尼 有弹性 粒子。