Extreme and Singular Behavior in Fundamental Models of Fluid Mechanics

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

Investigators

Abstract

This mathematical research project will develop and apply theoretical and computational tools to systematically search for extreme behavior in solutions of some of the fundamental equations of fluid mechanics. The goal is to derive precise predictions of physically significant quantities from first principles and understand the underlying nonlinear dynamical mechanisms. The work capitalizes on recent developments of the Principal Investigator, collaborators, and other scientists to implement ideas from optimal control theory and the calculus of variations to find flows achieving maximal mixing, optimal transport, or other extreme dissipation. Transport, mixing, and dissipation are among the most fundamental features of fluid flows and they are of foundational significance for important applications ranging from microfluidics engineering to climate science and astrophysics. The control and optimization techniques adopted here constitute novel computationally aided analysis approaches to these problems. This project directly involves advanced training for graduate students and postdoctoral researchers under the Principal Investigator?s direction at the University of Michigan. The research is centered on studies of qualitative and quantitative properties of solutions of partial differential equations in fluid mechanics including the Navier-Stokes and advection-diffusion equations. The investigator employs modern applied analysis and scientific computation in collaboration with graduate students and postdoctoral researchers at the University of Michigan, and with other researchers elsewhere. The four major components of the work are: (1) to study qualitative and quantitative properties of solutions to advection-diffusion equations and determine limits on mixing in terms of features of the stirring flows with an ultimate goal of understanding mixing effectiveness of turbulence; (2) to utilize new optimization schemes to investigate extreme time averages in nonlinear dynamical systems and investigate their implications for classical models of Rayleigh-B?nard convection; (3) to implement computational optimal control methods to search for maximal enstrophy production in the three-dimensional Navier-Stokes equations; and (4) to apply the analysis and computation techniques to other nonlinear and/or stochastic dynamical models from interdisciplinary applications in biology, chemistry and physics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.