Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

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

Investigators

Abstract

The investigator develops and applies effective mathematical analysis and scientific computation tools to systematically search for extreme behavior in some of the fundamental equations of physical fluid mechanics. The goal is to derive precise predictions of physically significant quantities from first principles. The work capitalizes on recent developments to implement ideas from optimal control theory and the calculus of variations to compute fluid flows achieving maximal mixing, optimal transport, or other extreme dissipation. Transport, mixing, and dissipation are among the most fundamental features of fluid flows and are of foundational significance for important applications ranging from microfluidics engineering to modeling in climate science and astrophysics. The control and optimization techniques adopted here constitute a new and unified computationally aided analysis approach to these problems. This project directly involves advanced training for graduate students and postdoctoral researchers. This project utilizes methods of modern applied mathematics and scientific computation. Mathematical measures of mixing introduced by the investigator and collaborators are utilized in optimal control analyses of the advection and advection-diffusion equations in order to place absolute limits on passive tracer mixing by incompressible flows, and to illuminate key features of particularly effective stirring strategies. Computational control and applied analysis are employed to construct incompressible fluid flows optimizing transport between impenetrable surfaces and produce new transport bounds for buoyancy-driven Rayleigh-Benard convection and the outstanding problem of turbulent convection. Optimal control techniques are developed and deployed to determine maximal enstrophy production in the incompressible three-dimensional Navier-Stokes equations over finite time intervals. Extremal solutions provide new insight into fully nonlinear vorticity amplification in unforced flows, and this component of the project is a novel and promising framework for the study of one of the signal challenges for 21st century applied mathematics: the regularity question for the 3D Navier-Stokes equations.