Strongly Nonlinear Wave and Transport Models in Stratified Fluids

波动是自然界最常见的现象之一，也是应用数学的核心课题。这一建议侧重于流体力学中的两个领域：海洋和大气中遇到的内部重力波，以及由大粘度对比度主导的两相界面上的波，这一设置发生在肺气道中。虽然这两个区域在相关的尺度上相距甚远，但它们共享共同的数学工具来设计能够描述和预测其主要动力学行为的模型。在第一个主题中，一个长期目标是提供一个分层分层中的内波模型，该模型正确地考虑了色散和高度非线性。其动机来自于越来越多的实验和现场证据，即大幅度内波是很容易获得的，也是海洋动力学的共同特征。第二个主题的重点是发展适当的模型，能够检测和跟踪一类核-环空流动中不稳定性的演变。其动机来自于对控制呼吸系统中粘液-空气流动耦合的流体动力学反馈机制的理解。利用动力学的长波性质，可以对这两个问题进行仔细的渐近分析，重点是强烈的非线性运动。现场和实验室数据检验了所得到的模型及其预测能力。该项目的总体目标是分离、理解特定的波动力学机制，并将其整合到工作模型中，实验、现场观察和分析表明，这些机制对于准确描述某些流体流动问题非常重要。这种方法的结果为实验和基准测试以及更精细的数值模拟提供了信息。通过新模型获得的理解对环境和健康科学产生了影响，例如，通过提高营养物质和污染物在环境中扩散的模拟的准确性，或者，对于该项目的第二部分，通过帮助制定改进的药物输送战略，非人道的呼吸道。

Wave motion is one of the most common phenomena of thenatural world and a subject central to Applied Mathematics. Thisproposal focuses on two areas in fluid mechanics: internal gravitywaves, as encountered in the ocean and atmosphere, and waves atthe interface between two phases dominated by a large viscositycontrast, a setup that occurs in the lung airways. While situatedfar apart in their relevant scales, these two areas nonethelessshare common mathematical tools for devising models able todescribe and predict their main dynamical behaviors. Within thefirst theme, a long term goal is to provide a model of internalwaves in layered stratification that correctly accounts fordispersion and high nonlinearity. The motivation comes from thegrowing body of evidence, experimental and in the field, thatlarge amplitude internal waves are easily attainable and a commonfeature of ocean dynamics. The focus of the second theme is todevelop the proper model capable of detecting and following theevolution of instabilities in a class of core-annular flows in apipe. The motivation comes from understanding the hydrodynamicfeedback mechanisms that govern mucus-air flow coupling inrespiratory systems. Careful asymptotic analysis taking advantageof the long wave nature of the dynamics can be brought to bear onboth problems, with emphasis on strongly nonlinear motion. Fieldand laboratory data test the resulting models and their predictivecapabilities. The overall aim of this project is to isolate, understand,and integrate into working models particular mechanisms of wavedynamics that experiments, field observations, and analysissuggest are important for an accurate mathematical description ofcertain fluid flow problems. The outcome of this approach informsand benchmarks experiments as well as more elaborate numericalsimulations. The understanding made available through the newmodels has an impact in environmental and health sciences, e.g.,by enhancing the accuracy of simulations of nutrient and pollutantdispersion in the environment, or, for the second part of thisproject, by helping devise improved strategies of drug delivery inhuman airways.