Well-posed and ill-posed behaviour of the µ ( I ) -rheology for granular flow
Well-posed and ill-posed behaviour of the µ ( I ) -rheology for granular flow
复制标题
颗粒流 µ ( I ) 流变学的适定和不适定行为
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
J. Gray
中科院分区：
文献类型：
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作者：
T. Barker;D. G. Schaeffer;P. Bohorquez;J. Gray
所有的物理系统都会受到噪声的影响，不会发生灾难性的爆炸。因此，寻求适定方程来作出切合实际的预测是至关重要的。最近的µ(I) -流变学是一个重要的进步，它可以预测溜槽和剪切细胞中的颗粒流动。这是通过在摩擦系数µ中引入对无量纲惯性数I的依赖来实现的。本文证明了μ (I) -流变学对I的中间值是适定的，但对高惯性数和低惯性数都是不适定的。从对方程的随意检查来看，这个结果并不明显，并且表明额外的物理，如持久力链和二元碰撞，在这些极限中变得重要。用两种隐式格式对非牛顿流体进行了数值验证。特别是，明确地表明，在给定的分辨率下，用于计算定常均匀巴格诺流的标准数值格式在参数空间的适定区域是稳定的，但在不适定区域对小的扰动是不稳定的，这些扰动呈指数级快速增长。
all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent µ( I ) -rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number I in the friction coefficient µ . In this paper it is shown that the µ( I ) -rheology is well-posed for intermediate values of I , but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain.