This proposal discusses the use of nonlinear dynamical systems theory in solving several long-standing problems of fluid mixing and flow separation. On the mathematical side, it will advance the theory of nonhyperbolic invariant manifolds, inertial manifolds, advection-diffusion and dynamo equations, finite-time invariant sets, stability of nonautonomous systems, and aperiodic averaging. On the fluid mechanical side, the proposed work will broaden the understanding of three-dimensional unsteady fluid separation, diffusive tracer mixing, the topology of three-dimensional vortices, and the origin of dynamo action in the induction equations.
In a broader sense, the proposal describes new approaches to nonlinear dynamical phenomena in oceanic and aerodynamic flows. Such phenomena have traditionally been poorly understood, because advanced mathematical theories, notably dynamical systems theory, have been underutilized in their study. The investigator proposes new theoretical tools for describing how different substances mix, and how high-speed fluid separates from boundaries. The proposal discusses how the results can improve ocean feature detection and prediction, advance aerodynamic design and aviation technology, and impact the design of efficient chemical mixers.
