Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India.
Phys Rev E. 2018 Jan;97(1-1):012902. doi: 10.1103/PhysRevE.97.012902.
Dense granular flows have been well described by the Bagnold rheology, even when the particles are in the multibody contact regime and the coordination number is greater than 1. This is surprising, because the Bagnold law should be applicable only in the instantaneous collision regime, where the time between collisions is much larger than the period of a collision. Here, the effect of particle stiffness on rheology is examined. It is found that there is a rheological threshold between a particle stiffness of 10^{4}-10^{5} for the linear contact model and 10^{5}-10^{6} for the Hertzian contact model above which Bagnold rheology (stress proportional to square of the strain rate) is valid and below which there is a power-law rheology, where all components of the stress and the granular temperature are proportional to a power of the strain rate that is less then 2. The system is in the multibody contact regime at the rheological threshold. However, the contact energy per particle is less than the kinetic energy per particle above the rheological threshold, and it becomes larger than the kinetic energy per particle below the rheological threshold. The distribution functions for the interparticle forces and contact energies are also analyzed. The distribution functions are invariant with height, but they do depend on the contact model. The contact energy distribution functions are well fitted by Gamma distributions. There is a transition in the shape of the distribution function as the particle stiffness is decreased from 10^{7} to 10^{6} for the linear model and 10^{8} to 10^{7} for the Hertzian model, when the contact number exceeds 1. Thus, the transition in the distribution function correlates to the contact regime threshold from the binary to multibody contact regime, and is clearly different from the rheological threshold. An order-disorder transition has recently been reported in dense granular flows. The Bagnold rheology applies for both the ordered and disordered states, even though the rheological constants differ by orders of magnitude. The effect of particle stiffness on the order-disorder transition is examined here. It is found that when the particle stiffness is above the rheological threshold, there is an order-disorder transition as the base roughness is increased. The order-disorder transition disappears after the crossover to the soft-particle regime when the particle stiffness is decreased below the rheological threshold, indicating that the transition is a hard-particle phenomenon.
密集颗粒流可以很好地用 Bagnold 流变学来描述，即使颗粒处于多体接触状态，配位数大于 1 也是如此。这令人惊讶，因为 Bagnold 定律应该只适用于瞬时碰撞状态，其中两次碰撞之间的时间远大于碰撞周期。在这里，研究了颗粒刚度对流变学的影响。发现对于线性接触模型，当颗粒刚度为 10^{4}-10^{5}时，对于赫兹接触模型，当颗粒刚度为 10^{5}-10^{6}时，存在流变学阈值，在该阈值之上，Bagnold 流变学（应力与应变速率的平方成正比）有效，而在该阈值之下，存在幂律流变学，其中所有应力和颗粒温度分量都与应变率的幂次成正比，该幂次小于 2。在流变学阈值处，系统处于多体接触状态。然而，在流变学阈值之上，每个颗粒的接触能量小于每个颗粒的动能，而在流变学阈值之下，每个颗粒的接触能量大于每个颗粒的动能。还分析了颗粒间力和接触能量的分布函数。分布函数与高度无关，但取决于接触模型。通过伽马分布很好地拟合接触能量分布函数。当线性模型的颗粒刚度从 10^{7}减小到 10^{6}，而赫兹模型的颗粒刚度从 10^{8}减小到 10^{7}时，当接触数超过 1 时，分布函数的形状会发生变化。因此，分布函数的变化与从二元接触状态到多体接触状态的接触状态阈值相关，并且明显不同于流变学阈值。最近在密集颗粒流中报道了一种有序-无序转变。即使流变学常数相差几个数量级，Bagnold 流变学也适用于有序态和无序态。在这里研究了颗粒刚度对有序-无序转变的影响。发现当颗粒刚度高于流变学阈值时，随着基底粗糙度的增加会发生有序-无序转变。当颗粒刚度低于流变学阈值时，在过渡到软颗粒状态后，有序-无序转变消失，表明该转变是一种硬颗粒现象。
