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.
