Strongly Nonlinear Wave and Transport Models in Stratified Fluids

波动是自然界中最常见的现象之一，也是应用数学的中心课题。 这一建议集中在流体力学的两个领域：内部重力波，如在海洋和大气中遇到的，和波在两相之间的界面由一个大的viscositycontrast占主导地位，一个设置，发生在肺气道。 虽然这两个领域在相关尺度上相距甚远，但它们却拥有共同的数学工具，用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，一个长期的目标是提供一个模型的internalwaves在分层分层，正确的帐户fordispersion和高非线性。 其动机来自于越来越多的实验和现场证据，即大振幅内波很容易获得，是海洋动力学的一个共同特征。 第二个主题的重点是发展适当的模型，能够检测和跟踪的不稳定性的发展，在一类核-环流动的apipe。 其动机来自于对呼吸系统中控制粘液-气流耦合的流体动力学反馈机制的理解。 仔细的渐近分析采取的长波性质的动力学可以带来承担onboth问题，强调强非线性运动。 现场和实验室数据测试所产生的模型和他们的预测能力。 这个项目的总体目标是分离，理解，并整合到工作模型的特定机制的波动力学实验，现场观察和分析表明，是重要的精确数学描述ofcertain流体流动问题。 这种方法的结果通知和基准实验以及更详细的数值模拟。 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，通过提高环境中营养物和污染物扩散模拟的准确性，或者，对于该项目的第二部分，通过帮助设计改进的药物输送非人气道策略。

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.