Measures of mixing in the blinking vortex map

Measures of mixing for fluid flows offer the practical application of knowing when/where a substance is mixed, and with a robust definition of mixing, provide a measure of chaos. When associating well-mixed regions with regions of chaotic motion, a measure of mixing can also be though of as a measure of chaos, and provide useful insight into the dynamics of the fluid flow. Here, two methods of mixing will be defined: (1) the Finite Time Lyapunov Exponent (FTLE) Field, and (2) the trace of the Moment of Inertia Tensor of the system's Density Distribution.;The FTLE field provides an estimate of the Lyapunov Exponents for a fixed grid of points[3]. A large Lyapunov Exponent represents motion where neighboring points move away from each other faster than points with lower Lyapunov exponents. Therefore, regions of the FTLE field that are large can be considered well mixed relative to the regions of the FTLE with lower values.;The second measure uses the density distribution of the flow. The density distribution is determined by dividing the domain into "cells" of equal area and allowing the flow to advect a single initial point "many" times. By summing up the number of times the point lands in a given cell, each will have a number that identifies how often the initial point "visited". This value can also be thought of as a density, particles per area, resulting in a density distribution over the domain. Furthermore, if the initial point were imagined to have a mass of one, each cell of the density distribution would contribute to the moment of inertia of the domain. The moment of inertia is quantified in a tensor, where the trace indicates whether the initial particle belongs to a mixed or unmixed region.;The "Blinking Vortex" Map, originally developed by Hassan Aref [1], is a model of stirring by chaotic advection in which diffusive mixing is neglected. Because this map is a stirring model, application of a measure of mixing will distinguish between stirred and unstirred regions of the flow.