Strongly Nonlinear Wave and Transport Models in Stratified Fluids

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

Wave motion is one of the most common phenomena of the natural world and a subject central to Applied Mathematics. This proposal focuses on two areas in fluid mechanics: internal gravity waves, as encountered in the ocean and atmosphere, and waves at the interface between two phases dominated by a large viscosity contrast, a setup that occurs in the lung airways. While situated far apart in their relevant scales, these two areas nonetheless share common mathematical tools for devising models able to describe and predict their main dynamical behaviors. Within the first theme, a long term goal is to provide a model of internal waves in layered stratification that correctly accounts for dispersion and high nonlinearity. The motivation comes from the growing body of evidence, experimental and in the field, that large amplitude internal waves are easily attainable and a common feature of ocean dynamics. The focus of the second theme is to develop the proper model capable of detecting and following the evolution of instabilities in a class of core-annular flows in a pipe. The motivation comes from understanding the hydrodynamic feedback mechanisms that govern mucus-air flow coupling in respiratory systems. Careful asymptotic analysis taking advantage of the long wave nature of the dynamics can be brought to bear on both problems, with emphasis on strongly nonlinear motion. Field and laboratory data test the resulting models and their predictive capabilities. The overall aim of this project is to isolate, understand, and integrate into working models particular mechanisms of wave dynamics that experiments, field observations, and analysis suggest are important for an accurate mathematical description of certain fluid flow problems. The outcome of this approach informs and benchmarks experiments as well as more elaborate numerical simulations. The understanding made available through the new models has an impact in environmental and health sciences, e.g., by enhancing the accuracy of simulations of nutrient and pollutant dispersion in the environment, or, for the second part of this project, by helping devise improved strategies of drug delivery in human airways.