刚性颗粒在Bingham流体和Cheeger集中沉降的临界屈服数
Critical yield numbers of rigid particles settling in Bingham fluids and Cheeger sets
摘要：
本文研究了在宾汉流体边界区域内，致密固体夹杂物（颗粒）在重力作用下沉降的临界屈服数$Y_c$，即足以阻止运动的屈服应力与浮力应力的临界比值。我们把自己限制在一个二维平面配置与一个单一的反平面速度分量。因此，粒子域和流体域都是具有固定横截面的无限圆柱体。我们表明，这样的屈服数产生于一个本征值问题的约束全变差。通过连续求解两个Cheeger型集最优化问题，构造了该问题的特解。我们提出了一些示例几何中，这些几何解决方案可以明确地找到，并讨论解决方案的一般功能。最后，我们考虑一个计算方法的本征值问题，这是在数值实验中看到产生这些几何解决方案。
英文摘要：
We consider the fluid mechanical problem of identifying the critical yield number $Y_c$ of a dense solid inclusion (particle) settling under gravity within a bounded domain of Bingham fluid, i.e. the critical ratio of yield stress to buoyancy stress that is sufficient to prevent motion. We restrict ourselves to a two-dimensional planar configuration with a single anti-plane component of velocity. Thus, both particle and fluid domains are infinite cylinders of fixed cross-section. We show that such yield numbers arise from an eigenvalue problem for a constrained total variation. We construct particular solutions to this problem by consecutively solving two Cheeger-type set optimization problems. We present a number of example geometries in which these geometric solutions can be found explicitly and discuss general features of the solutions. Finally, we consider a computational method for the eigenvalue problem, which is seen in numerical experiments to produce these geometric solutions.
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