9404554 Shelley Large-scale computation and computational mathematics are now primary tools for studying those physical processes characterized by randomness and strong nonlinearity. These processes include complex pattern formation and growth problems in diffusion dominated systems, and wave interactions on the ocean's free surface. Closely related are problems with singular or nearly singular behavior, such as flows undergoing changes in topological type, or water wave breaking. The main objective of this proposal is to investigate such problems in fluids and materials by developing and applying state-of-the-art numerical methods to large-scale computation, and understanding important constituent processes through analysis and modelling. All these problems, while arising in quite disparate physical settings, are strongly mediated by the presence of surface tension or energy, and often share mathematical structure. 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. One such problem is understanding pattern formation in diffusion dominated systems. This has important implications in the production of metallic alloys. The complex patterns that are observed to form within the alloys during their production, strongly affect the bulk properties of the resulting alloys, such as their strength and toughness. Other problems involve the motions of free surfaces in fluid flows. Two important instances are the interactions of waves on the ocean's surface, and the dynamics of flows undergoing changes in topological type, say as characterized by the production of bubbles through the mixing of two fluids. All of the above problems are similar in the sense that the relevant phenomena is heavily influenced by surface tension at the interface (called "surface energy" in the materials science context). As such, they c an be studied by some common analytical and computational tools.
