Topological Characterization and Optimization of Mixing

混合的拓扑表征和优化

• 批准号: 0806821
• 负责人: Jean-Luc Thiffeault
• 金额: $ 20万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806821&HistoricalAwards=false
• 关键词: Topological; Characterization; Optimization; Mixing; Topological; Characterization; Optimization; Mixing

Many industrial applications require the mixing or blending of ingredients and reagents, often at great energetic costs. Examples include food processing, polymer processing, and microfluidics.Stirring and mixing are also crucial ingredients for quantifying climate change. Most approaches to these problems involve direct solution of the underlying differential equations. In this proposal we advocate a complementary viewpoint that focuses on the topology of stirring. For the case of stirring with rods, topology refers to the orbits that the rods follow, and how these orbits interlock. Our two broad aims are the characterization and optimization of stirring devices from a topological viewpoint. To characterize means to describe existing devices or stirring configurations using a topological approach, in order to learn as much as possible about the device without necessarily solving the full governing fluid equations. A thorough characterization allows us to then optimize devices to maximize their topological mixing characteristics. Our research applies a powerful body of existing mathematics -- the Thurston-Nielsen theory of surface transformations -- to fluid-dynamical problems. This theory has not yet been fully exploited in practical situations. Since the topological approach does not use differential equations directly, many of the results are independent of the specific fluid involved. One example of the type of optimization made possible is a class of rod-stirring devices we call the silver mixers: their "topological entropy" is a multiple of the silver ratio, the golden ratio's lesser-known cousin. Silver mixers are optimal in the sense that there is no motion of rods that can "stretch" so-called material lines at a faster rate.Many industrial applications require the mixing or blending of ingredients and reagents, often at great energetic costs. Examples include food processing, polymer processing, and microfluidics -- the emerging science of manipulating fluids at microscopic and nanoscales. Stirring and mixing are also crucial ingredients for quantifying climate change, for example to estimate the rate of absorption of greenhouse gases by the ocean. In all cases, it is important to understand the nature of the mixing process in order to improve it, thereby increasing quality or yield. Most approaches to these problems involve direct solution of the underlying equations, a formidable task even with today's computers. We advocate a complementary viewpoint that focuses on geometrical features of stirring -- or "topological" in mathematical language. For the case of stirring with rods, topology refers to the orbits that the rods follow, and how these orbits interlock. Recent developments in mathematics provide us with novel tools to study this problem. Our two broad aims are the characterization and optimization of stirring devices from a topological viewpoint. To characterize means to describe existing devices or stirring configurations using a topological approach, in order to learn as much as possible about the device without necessarily solving the full equations. A thorough characterization allows us to then optimize devices to maximize their topological mixing characteristics. For example, given a small number of rods constrained to move on gears, a typical situation in the food industry, we have devised a device that leads to optimal mixing merely by tweaking the trajectory of individual rods.

许多工业应用都需要将成分和试剂的混合或混合，通常是以巨大的成本。例子包括食物加工，聚合物加工和微流体。结合和混合也是量化气候变化的关键成分。解决这些问题的大多数方法都涉及基础微分方程的直接解决方案。在此提案中，我们主张一个互补的观点，重点是搅动的拓扑。对于与杆搅拌的情况，拓扑是指杆遵循的轨道，以及这些轨道如何互锁。我们的两个广泛目的是从拓扑观点来表征和优化搅拌设备。为了描述现有设备或使用拓扑方法搅动配置的方法，以便尽可能多地了解该设备，而无需求解完整的管理流体方程。彻底的表征使我们可以优化设备以最大化其拓扑混合特性。我们的研究将现有数学的强大机构（Thurston-Nielsen的表面转化理论）应用于流体动力问题。在实际情况下，该理论尚未得到充分利用。由于拓扑方法不直接使用微分方程，因此许多结果与所涉及的特定流体无关。优化类型的一个例子是我们称为银色混合器的一类杆 - 刺激器件：它们的“拓扑熵”是银比的倍数，即黄金比率鲜为人知的堂兄。 银色混合器是最佳的，因为没有杆的运动能够以更快的速度“伸展”所谓的材料线。许多工业应用需要混合或混合成分和试剂，通常以巨大的能量成本。例子包括食物加工，聚合物加工和微流体学 - 在微观和纳米级操纵液体的新兴科学。搅拌和混合也是量化气候变化的关键成分，例如估计海洋吸收温室气体的速度。在所有情况下，重要的是要了解混合过程的性质以改进它，从而提高质量或产量。解决这些问题的大多数方法都涉及基础方程的直接解决方案，即使是当今计算机，也是一项艰巨的任务。我们提倡一个互补的观点，该观点重点介绍了搅拌的几何特征 - 或数学语言中的“拓扑”。对于与杆搅拌的情况，拓扑是指杆遵循的轨道，以及这些轨道如何互锁。 数学方面的最新发展为我们提供了研究此问题的新工具。我们的两个广泛目的是从拓扑观点来表征和优化搅拌设备。为了描述现有设备或使用拓扑方法搅拌配置的方法，以便在不必求解完整方程的情况下尽可能多地了解该设备。彻底的表征使我们可以优化设备以最大化其拓扑混合特性。例如，鉴于少量的杆被限制在齿轮上移动，这是食品行业的典型情况，我们设计了一种设备，该设备仅通过调整单个杆的轨迹而导致最佳混合。