Applied Analysis of the Navier-Stokes and Related Equations

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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.