Computations and Analysis of Fluids and Materials

本提案的主要目标是通过(1)开发和应用最先进的数值方法进行大规模计算，(2)对重要组成过程进行分析、数值和建模研究，来研究流体-流体和固体-固体界面的动力学。具体来说，重点将是研究流体中的拓扑转变和固体中微观结构的扩散演化。这些领域涉及基本的物理过程，其现象学是理解真实流体行为和固体材料性质的基础。两者的特点都是存在多个组成成分，复杂的模式形成和/或奇点（即空间复杂性）。尽管这些过程出现在非常不同的物理现象（流体与固体）中，但都涉及自由边界问题，其中分离不同组分的边界界面的运动是由表面能与不稳定性或多体相互作用之间的竞争驱动的。因此，它们可以使用一组通用的分析和计算工具来处理。这些问题的高度非线性性质使得快速、准确和鲁棒的数值方法对它们的研究至关重要。在本提案中，我们将数学和数值分析，建模和大规模科学计算结合起来研究流体动力学和材料科学中的某些基本问题。例如，我们将考虑的一个问题涉及不同流体之间界面拓扑结构的变化。例如，当液体射流挤压成液滴时，当一种液体的液滴与另一种液体的液滴重新连接时，就会发生这种变化。这些拓扑转换发生在许多实际应用中，包括石油、化工和食品产品的运输、混合和分离，以及石油泄漏等环境应用。通常，这些系统中的反应和混合速率直接由过渡过程的详细动力学控制。因此，有必要了解这些动力学，以便为混合和反应速率预测开发准确的工程模型。我们将使用分析、建模和大规模科学计算来研究流体界面破裂和重新连接的详细动力学。我们要考虑的另一个问题涉及固态扩散相变。这些转化是加工多组分金属合金（如钢）的重要方法。这一过程的结果是形成多相组织，这是设定合金宏观力学性能（即刚度、强度和韧性）的关键变量。微观结构的特点是不同金属成分的区域被界面分开。我们的研究目标是准确地模拟和模拟合金中微观结构的形成，以便为冶金学家提供生产具有理想材料性能的新合金的配方。尽管上面描述的两个问题产生于非常不同的物理过程（流体与固体），但它们在相关现象受到各自界面表面张力的强烈影响的意义上是相似的。因此，它们可以使用常见的分析和计算工具进行研究。这些问题的高度复杂性使得快速、准确和稳健的数值方法对它们的研究至关重要。

9706931 Lowengrub The main objective of this proposal is to investigate the dynamics of fluid-fluid and solid-solid interfaces by (1) developing and applying state-of-the-art numerical methods to large scale computation and (2) performing analytical, numerical and modelling studies of important constituent processes. Specifically, the focus will be on studying topological transitions in fluids and the diffusional evolution of microstructure in solids. These areas involve fundamental physical processes whose phenomenology is basic to understanding the behavior of real fluids and the material properties of solids. Both are characterized by the presence of multiple constitutive components, complex pattern formation and/or singularities (i.e. spatial complexity). Although these processes arise in very different physical phenomena (fluids versus solids), both involve free boundary problems where the motion of a bounding interface, separating the different components, is driven by a competition between surface energy and either an instability or multi-body interactions. As such, they can be treated using a common set of analytical and computational tools. The highly nonlinear nature of these problems makes fast, accurate and robust numerical methods essential to their study. In this proposal, we bring together mathematical and numerical analysis, modelling, and large-scale scientific computation to study certain fundamental problems in fluid dynamics and materials science. For instance, one problem we will consider concerns changes in the topology of interfaces between different fluids. Such changes occur, for example, when liquid jets pinch off into droplets and when droplets of one fluid reconnect with another. These topological transitions occur in many practical applications involving transport, mixing, and separation of petroleum, chemical, and food products as well as in environmental applications such as oil spills. Often, reaction and mixing rates within these systems are controlled directly by the detailed dynamics of the transition processes. Thus, there is a need to understand these dynamics in order to develop accurate engineering models for mixing and reaction rate prediction. We will use analysis, modelling and large scale scientific computation to investigate the detailed dynamics of break-up and reconnection of fluid interfaces. Another problem we will consider involves solid-state diffusional phase transformations. These transformations are an important method of processing multicomponent metallic alloys such such as steels. The result of this process is the formation of a multiphase microstructure, which is a key variable in setting the macroscopic mechanical properties (i.e. stiffness, strength and toughness) of the alloy. The microstructure is characterized by regions of different metallic components separated from one another by interfaces. The goal of our research is to accurately model and simulate the formation of microstructure in alloys in order to provide metallurgists with a recipe for generating new alloys with desirable material properties. Although the two problems described above arise from very different physical processes (fluids versus solids), they are similar in the sense that the relevant phenomena is strongly influenced by surface tension at the respective interfaces. Consequently, they can be studied using common analytical and computational tools. The highly complex nature of these problems makes fast, accurate and robust numerical methods essential to their study.