Date of Award

Spring 1-1-2017

Document Type


Degree Name

Doctor of Philosophy (PhD)


Applied Mathematics

First Advisor

James D. Meiss

Second Advisor

Keith Julien

Third Advisor

Juan Restrepo

Fourth Advisor

John Crimaldi

Fifth Advisor

Roseanna Neupauer


Mixing of a passive scalar in a fluid flow results from a two part process in which large gradients are first created by advection and then smoothed by diffusion. We investigate methods of designing efficient stirrers to optimize mixing of a passive scalar in a two-dimensional nonautonomous, incompressible flow over a finite time interval. The flow is modeled by a sequence of area-preserving maps whose parameters change in time, defining a mixing protocol. Stirring efficiency is measured by the mix norm, a negative Sobolev seminorm; its decrease implies creation of fine-scale structure. A Perron-Frobenius operator is used to numerically advect the scalar for three examples: compositions of Chirikov standard maps, of Harper maps, and of blinking vortex maps. In the case of the standard maps, we find that a protocol corresponding to a single vertical shear composed with horizontal shearing at all other steps is nearly optimal. For the Harper maps, we devise a predictive, one-step scheme to choose appropriate fixed point stabilities and to control the Fourier spectrum evolution to obtain a near optimal protocol. For the blinking vortex model, we devise two schemes: A one-step predictive scheme to determine a vortex location, which has modest success in producing an efficient stirring protocol, and a scheme that finds the true optimal choice of vortex positions and directions of rotation given four possible fixed vortex locations. The results from the numerical experiments suggest that an effective stirring protocol must include not only steps devoted to decreasing the mix norm, but also steps devoted to preparing the density profile for future steps of mixing.