DynSyst_Special_Topics: A Dynamical Systems Approach to Mixing and Segregation of Granular Matter

颗粒混合及其与偏析的相互作用是复杂的，因为流动会通过颗粒尺寸或密度引起偏析。 研究颗粒流的一个典型系统是部分充满的旋转颗粒流。 因为，在合理的近似下，湍流中的动力学发生在薄的流动表面层中，所以可以使用简单、紧凑和可扩展的连续介质模型来研究流动。 然而，一个合适的混合（和隔离）以及适当的数学工具来实现理论的一般理论框架缺乏。 充分理解需要综合努力，包括由抽象数学概念启发的新理论和支持计算和实验结果。 这里提出的理论工作将集中在应用程序的分段等距，链接扭曲映射，拉格朗日相干结构。 研究的最终目标是开发一个理论框架，使用动力系统方法，将导致新的数学工具来预测流动，混合，分离，和图案形成的颗粒物质在2D和3D几何形状与稳定和时间周期的强迫。 该方法是基于由强制（混合协议）产生的三维流的几何形状和对称性。 补充实验和模拟将被用来确认这些理论方法的适用性，真实的颗粒混合和离析问题。颗粒物质流动的物理学是科学中的一个大问题。 颗粒物质是一个复杂系统的原型，其集体行为远离平衡。 然而，许多基本问题仍然存在。 同时，对颗粒流的理解在从滑坡到食品加工的各种情况下具有巨大的实际意义。流动颗粒系统是强烈无序的，但显示混乱（混合）和秩序（分离）之间的竞争。 但迄今为止的进展一直不大。 在这里，我们介绍了新的动力系统方法的颗粒物质的研究是基于高等数学。动力系统工具提供了可以在颗粒流研究中利用的数学框架。这里的科学进步可以对技术和实践产生直接影响。此外，这里考虑的新的数学方法有可能对许多物理系统产生广泛的影响，也许会创造一个全新的范式，就像20世纪90年代混沌平流科学对流体混合所做的那样。

Granular mixing and its interplay with segregation are complicated, since flow induces segregation by particle size or density. A canonical system for the study of granular flow is a partially filled rotating tumbler. Because, to a reasonable approximation, the dynamics in a tumbler take place in a thin flowing surface layer, a simple, compact, and extensible continuum model can be used to study the flow. However, a suitable general theoretical framework for mixing (and segregation) as well as appropriate mathematical tools to implement the theory are lacking. Full understanding requires an integrated effort consisting of new theory inspired by abstract mathematical concepts and supporting computational and experimental results. The theoretical work proposed here will focus on the application of Piecewise Isometries, Linked Twist Maps, and Lagrangian Coherent Structures. The ultimate objective of the research is to develop a theoretical framework using a dynamical systems approach that will lead to new mathematical tools to predict flow, mixing, segregation, and pattern formation for granular matter in 2d and 3d tumbler geometries with steady and time-periodic forcing. The approach is based on the geometry and symmetries of the 3d flow generated by the forcing (mixing protocols). Complementary experiments and simulations will be used to confirm the applicability of these theoretical approaches to real granular mixing and segregation problems. The physics of the flow of granular matter is one of the big questions in science. Granular matter is a prototype of a complex system with collective behavior far from equilibrium. Yet many fundamental questions remain. At the same time, an understanding of granular flow has tremendous practical importance in situations ranging from landslides to food processing. Flowing granular systems are strongly disordered and yet display competition between chaos (mixing) and order (segregation). But inroads to date have been modest. Here, we introduce new dynamical systems approaches for the study of granular matter that are grounded in higher mathematics. Dynamical systems tools offer mathematical frameworks that can be exploited in the study of granular flow. Scientific progress here can have an immediate impact on technology and practice. Furthermore, the new mathematical approaches considered here have potential for broad-ranging impact on many physical systems, perhaps creating an entirely new paradigm much like the science of chaotic advection did for mixing of fluids in the 1990s.