Fundamental Mathematics of Boundary Interactions and Transport in Stratified Fluids

The motion of fluids in the presence of realistic boundary conditions, such as those generated by an immersed moving body coupled to the fluid by viscous and inertial forces, is arguably one of the most important challenges in the mathematical theory of fluids and a problem relevant to many different scientific disciplines, ranging over a wide variety of scales from the distribution ofenvironmental pollutants in the atmosphere and oceans to the feeding of the smallest micro-organism. The interplay between coupling body fluid forces may be further complicated by the presence of stratification in the fluid, through entrainment phenomena as well as enhanced mixing by the flowwhich in turn can create buoyancy-like effective forcing on the conglomerate system of body and fluid. Given the complexity of such systems, our proposal outlines a series of fundamental problems. Specifically, we propose a combined theoretical, numerical and experimental e -ort on three research areas, where bodies and coherent structures (such as internal waves in ocen settings)interact with confining boundaries. We stress that while the motivation for this research resides ultimately in applications such as geophysics, our emphasis is on the fundamental fluid mechanical understanding of specific effects at scales realizable in our fluids lab. Some of the novel mechanismswe propose to study may then be scaled up to field applications by identifying the appropriate parametric regimes. Our target areas include internal waves in stratified fluids and their interaction with top and bottom boundary confinement surfaces, including the case of sea-ice caps, diffusion driven aggregation of solid particles through a newly discovered force, and the role of geometry inflows near impact points of rigid bodies with confining boundaries.The environment in which we live and breathe is a complex coupled fluid system whose dynamics possess phenomena occurring on a vast range of space and time scales. From the smallest cilia in the lung which provide a hydrodynamic defense mechanism against inhaled contaminants tothe atmosphere, oceans, and our climate, we interact directly with our fluid environment. The complete description of such a system remains beyond the computational scope of even the largest supercomputers, and a fundamental scientific endeavor is to characterize, and understand phenomena on different scales, and ultimately to integrate these phenomena to offer a grand picture of how this large scale system functions, as a whole. With this in mind, this proposal invokes a program focusing upon fundamental phenomena explored through a combination of rigorous mathematical hydrodynamic theory and experiment which will improve our understanding of some aspects of ourcomplex environment and provide the foundation and validation for some of the direct numerical approaches pursued through computational modeling. The majority of this theoretical and experimental research will take place in and around the new UNC Joint Fluids Laboratory. This laboratory will provide an optimal avenue towards broad impact of the proposed research, boththrough its impact on problems relevant to marine dynamics, as well as through a successful undergraduate component and the training of graduate students. Further, the lab provides broad outreach to middle and high school students through tours and summer research projects. The proposal seeks to provide both improved scientific understanding, and along the way, great opportunities for research and educational experiences for undergraduate and graduate students of the natural sciences.