The stress tensor in a granular flow at high shear rates
The stress tensor in a granular flow at high shear rates
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DOI：
10.1017/s0022112081000736
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发表时间：
1981-09
影响因子：
3.7
通讯作者：
S. Savage;D. J. Jeffrey
中科院分区：
文献类型：
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作者：
S. Savage;D. J. Jeffrey
在颗粒剪切流中的应力张量的计算假设，组成颗粒质量的粒子之间的二元碰撞是负责大部分的动量输运。我们假设粒子是光滑的、坚硬的、弹性的球体，并将应力表示为包含粒子速度及其空间排列的概率分布函数的积分。通过假设单粒子速度分布函数是麦克斯韦分布函数，空间对分布函数是由一个公式，由于卡纳汉和Starling，我们减少了这个积分到一个依赖于一个单一的无量纲参数R：特征平均剪切速度的比的均方根碰撞前粒子速度扰动。该积分是渐近评估为R [Gt ] 1和R [Lt ] 1和数值为中间值。当R取1·7时，在干燥颗粒材料上的实验测得的应力与理论预测值之间有很好的一致性。这种情况可能是本分析最适合的情况。对于中等和大的R值，理论预测的剪切应力和法向应力是成比例的颗粒直径的平方和剪切速率的平方，并强烈依赖于固体的体积分数。本文对悬浮在水中的中性浮力蜡球的剪切流，在极限R → ∞时所预言的应力与Bagnold的实验结果作了初步的比较。预测的应力是正确的数量级，并产生适当的变化的应力与浓度。当R [Lt ] 1时，剪切应力与剪切速率成线性关系，该分析可应用于流化床中的剪切流，但在此不进一步发展这种应用。
The stress tensor in a granular shear flow is calculated by supposing that binary collisions between the particles comprising the granular mass are responsible for most of the momentum transport. We assume that the particles are smooth, hard, elastic spheres and express the stress as an integral containing probability distribution functions for the velocities of the particles and for their spatial arrangement. By assuming that the single-particle velocity distribution function is Maxwellian and that the spatial pair distribution function is given by a formula due to Carnahan & Starling, we reduce this integral to one depending upon a single non-dimensional parameter R: the ratio of the characteristic mean shear velocity to the root mean square of the precollisional particle-velocity perturbation. The integral is evaluated asymptotically for R [Gt ] 1 and R [Lt ] 1 and numerically for intermediate values. Good agreement is found between the stresses measured in experiments on dry granular materials and the theoretical predictions when R is given the value 1·7. This case is probably the one for which the present analysis is most appropriate. For moderate and large values of R, the theory predicts both shear and normal stresses that are proportional to the square of the particle diameter and the square of the shear rate, and depend strongly on the solids volume fraction. A provisional comparison is made between the stresses predicted in the limit R → ∞ and the experimental results of Bagnold for shear flow of neutrally buoyant wax spheres suspended in water. The predicted stresses are of the correct order of magnitude and yield the proper variation of stress with concentration. When R [Lt ] 1, the shear stress is linear in the shear rate, and the analysis can be applied to shear flow in a fluidized bed, although such an application is not developed further here.