Applied Analysis of the Navier-Stokes and Related Equations

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

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 of uid mechanics. The Navier-Stokes equations constitute the basic mathematical model of uid ow, and are believed to contain turbulent dynamics among their solutions. Turbulence in uid mechanics 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 sciences and engineering. The work in this project will be carried out via modern applied analysis and numerical computation and simulation by the principal investigator (PI), Prof. Charles R. Doering of the University of Michigan, and graduate students performing doctoral dissertation work. This project has three specific objectives: For one, a mathematical technique for deriving rigorous bounds on turbulent dissipation and drag, which has come to be known as the \background method," will be developed and expanded to new areas including magnetohydrodynamics, drag-reducing polymer ows, and ows over rough boundaries. The background method was introduced by the PI and his collaborator a decade ago for the Navier-Stokes equations, and since then it has been developed and applied by the PI, his students, and many other researchers to a number of fundamental shear ow and thermal convection problems. One particular goal of this project will be to explore applications to a wider variety of problems 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 capable of putting limits on the heat transfer rate. Problems of concern here include laminar and turbulent convection with free-slip boundaries, xed-ux convection, ows driven by internal heating, and infinite Prandtl number models inspired by applications in geophysics. In a third direction of research, power consumption and enstrophy generation will be studied for forced ows and free ows in the absence of rigid boundaries. The PI and collaborators have recently 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 for unforced ows is to solve a variational problem for the maximum enstrophy-generating configuration and study how it relates to structures observed in fully developed turbulence or the potential development of singularities. With regard to the intellectual merit of this activity, knowledge gained from this project will further our understanding of some basic mathematical models in uid dynamics of direct relevance to many branches of engineering and applied science. In the long term, this kind of mathematical research could help the development of practical techniques for the prediction and/or control of physical 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 graduate students within the University of Michigan's new Ph.D. program in Applied & Interdisciplinary Mathematics. Moreover, this project also involves other investigators|including graduate students and postdoctoral researchers from the University of Michigan as well as other institutions|who will collaborate in the research.