粒子阵列在宾汉流体中沉降的临界屈服数和极限屈服面
Critical yield numbers and limiting yield surfaces of particle arrays settling in a Bingham fluid
摘要：
本文研究了反平面剪切流场中宾汉流体中多粒子的流动。限制的情况下，其中的内部和施加的力的平衡和流体和颗粒停止流动，也就是说，当流动稳定，被制定为寻找之间的最佳比率的全变差功能和线性泛函。这个商的极小值被称为临界屈服数，或者类似于瑞利屈服数，称为广义本征值。这个最小值一般只能通过不连续的速度而不是物理的速度来达到。然而，我们证明，这些广义本征函数，其跳跃，我们称之为限制屈服面，出现作为重新缩放的限制的物理速度。然后，我们证明了几何简单极小的存在性。此外，一个数值方法的最小化，然后考虑。它是基于一个非线性有限差分离散，其一致性证明，和一个标准的原始-对偶下降计划。最后，数值例子显示了各种几何解，表现出理论部分中讨论的性质。
英文摘要：
We consider the flow of multiple particles in a Bingham fluid in an anti-plane shear flow configuration. The limiting situation in which the internal and applied forces balance and the fluid and particles stop flowing, that is, when the flow settles, is formulated as finding the optimal ratio between the total variation functional and a linear functional. The minimal value for this quotient is referred to as the critical yield number or, in analogy to Rayleigh quotients, generalized eigenvalue. This minimum value can in general only be attained by discontinuous, hence not physical, velocities. However, we prove that these generalized eigenfunctions, whose jumps we refer to as limiting yield surfaces, appear as rescaled limits of the physical velocities. Then, we show the existence of geometrically simple minimizers. Furthermore, a numerical method for the minimization is then considered. It is based on a nonlinear finite difference discretization, whose consistency is proven, and a standard primal-dual descent scheme. Finally, numerical examples show a variety of geometric solutions exhibiting the properties discussed in the theoretical sections.
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