Applied Analysis of the Navier-Stokes and Related Equations

这是对数学物理和应用数学基础研究的一个建议，重点是不可压缩Navier-Stokes方程和相关的流体力学方程所提出的挑战。Navier-Stokes方程构成了水流的基本数学模型，并且被认为在其解中包含湍流动力学。流体力学中的湍流问题是理论物理和应用数学面临的突出挑战之一，在物理科学和工程的许多领域有着重要的应用和意义。本课题的主要研究者Charles R.密歇根大学的Doering，以及正在进行博士论文工作的研究生。这个项目有三个具体目标：首先，一个数学技术推导严格的边界湍流耗散和阻力，这已被称为“背景方法”，将被开发和扩展到新的领域，包括磁流体力学，减阻聚合物OWS，和OWS在粗糙边界。背景方法是由PI和他的合作者在十年前为Navier-Stokes方程引入的，从那时起，PI，他的学生和许多其他研究人员已经开发并应用于一些基本的剪切流和热对流问题。该项目的一个特别目标是探索应用于更广泛的科学问题，该项目的另一个重点是继续正在进行的研究，在热对流模型分析中的理论和数学问题，其中背景方法能够限制传热速率。这里所关心的问题包括自由滑移边界的层流对流和强迫对流、xed-ux对流、内部加热驱动的OWS以及受物理学应用启发的无限普朗特数模型。在第三个研究方向中，将研究没有刚性边界的强迫OWS和自由OWS的能量消耗和涡拟能产生。PI和合作者最近开发了一种新的方法来分析由时间无关的体力驱动的湍流，并建议将结果扩展到时间相关的力。非受迫湍流的一个独特问题是解决最大拟能生成构形的变分问题，并研究它与充分发展的湍流中观察到的结构或奇点的潜在发展之间的关系。从这个项目中获得的知识将进一步加深我们对流体动力学中与许多工程分支直接相关的一些基本数学模型的理解，并应用于科学从长远来看，这类科学研究可以帮助发展预测和/或控制从气象学到材料制造的物理过程的实用技术。就这一活动的广泛影响而言，该项目有几个重要的高级培训方面。首先，它为密歇根大学新博士学位的研究生提供研究支持和机会。应用跨学科数学专业。此外，该项目还涉及其他调查人员|包括来自密歇根大学和其他机构的研究生和博士后研究人员|who will collaborate合作in the research研究.

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.