Applied Analysis of the Navier-Stokes and Related Equations

纳维-斯托克斯及相关方程的应用分析

• 批准号: 0244859
• 负责人: Charles Doering
• 金额: $ 26.7万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244859&HistoricalAwards=false
• 关键词: Applied; Analysis; Navier; Stokes; Related

This is a proposal for fundamental research in mathematical physics and applied mathematics fo-cused on the challenges presented by the incompressible Navier-Stokes and related equations ofuid mechanics. The Navier-Stokes equations constitute the basic mathematical model of uidow, and are believed to contain turbulent dynamics among their solutions. Turbulence in uidmechanics remains one of the outstanding challenges for theoretical physics and applied mathe-matics with important applications in, and implications for, many areas in the physical sciencesand engineering. The work in this project will be carried out via modern applied analysis andnumerical computation and simulation by the principal investigator (PI), Prof. Charles R. Doeringof the University of Michigan, and graduate students performing doctoral dissertation work. Thisproject has three specific objectives:For one, a mathematical technique for deriving rigorous bounds on turbulent dissipation anddrag, which has come to be known as the \background method," will be developed and expandedto new areas including magnetohydrodynamics, drag-reducing polymer ows, and ows over roughboundaries. The background method was introduced by the PI and his collaborator a decade agofor the Navier-Stokes equations, and since then it has been developed and applied by the PI, hisstudents, and many other researchers to a number of fundamental shear ow and thermal convectionproblems. One particular goal of this project will be to explore applications to a wider variety ofproblems of scientific interest.Another focus of this project is to continue the ongoing investigation of theoretical and mathe-matical issues in the analysis of thermal convection models where the background method is capableof putting limits on the heat transfer rate. Problems of concern here include laminar and turbulentconvection with free-slip boundaries, xed-ux convection, ows driven by internal heating, andinfinite Prandtl number models inspired by applications in geophysics.In a third direction of research, power consumption and enstrophy generation will be studiedfor forced ows and free ows in the absence of rigid boundaries. The PI and collaborators haverecently developed a new approach for the analysis of turbulence driven by time-independent body-forces, and it is proposed to extend the results to time-dependent forces. A distinct problem forunforced ows is to solve a variational problem for the maximum enstrophy-generating configurationand study how it relates to structures observed in fully developed turbulence or the potentialdevelopment of singularities.With regard to the intellectual merit of this activity, knowledge gained from this project willfurther our understanding of some basic mathematical models in uid dynamics of direct relevanceto many branches of engineering and applied science. In the long term, this kind of mathematicalresearch could help the development of practical techniques for the prediction and/or control ofphysical processes ranging from meteorology to materials manufacturing.And with regard to this activity's broader impacts, there are several significant advanced train-ing aspects to the project. For one, it provides research support and opportunities for graduatestudents within the University of Michigan's new Ph.D. program in Applied & InterdisciplinaryMathematics. Moreover, this project also involves other investigators|including graduate studentsand postdoctoral researchers from the University of Michigan as well as other institutions|whowill collaborate in the research.

这是针对数学物理学基础研究的一项建议，并在不可压缩的Navier-Stokes和相关方程式机制所面临的挑战上应用了数学。 Navier-Stokes方程构成了Uidow的基本数学模型，并且据信它们的解决方案中包含湍流动力学。 UIDMechanics的湍流仍然是理论物理学和应用数学伴侣的重大挑战之一，具有重要的应用，并对物理科学和工程中的许多领域产生了影响。该项目的工作将通过现代应用分析和数量计算和仿真进行，由密歇根大学Charles R. Doering教授以及从事博士学位论文工作的研究生进行。该项目具有三个具体的目标：对于一个数学技术，用于在湍流耗散和drag上得出严格的界限，这已被称为\背景方法，”将开发并扩展到新领域，并扩展到新领域从那以后，PI，HISSTUDENS和许多其他研究人员对此进行了开发和应用。这里关注的问题的限制包括层状和自由滑动边界，Xed-ux对流，由内部供暖驱动的OWS，以及受到地球物理中应用程序启发的启发。 PI和合作者为分析由时间独立的身体强度驱动的湍流开发了一种新方法，并提议将结果扩展到时间依赖性力。一个明显的问题是解决一个变化问题，以最大程度地陷入困境的配置和研究如何与在完全发达的湍流或奇异性的实力发展中观察到的结构有关。与该项目的知识相关的知识将对某些基本的分支构建中的构图和相关的分支机构的理解，这是从该项目中获得的。从长远来看，这种数学研究可以帮助开发从气象学到材料制造的物理过程的预测和/或控制过程的实用技术。关于这项活动的更广泛的影响，该项目有几个重要的先进培训方面。首先，它为密歇根大学的新博士学位提供了研究支持和机会。应用和跨学科的计划。此外，该项目还涉及其他研究人员|包括密歇根大学的研究生和博士后研究人员以及其他机构|在研究中合作。