Mixing by Cutting and Shuffling a Line Segment: The Effect of Incorporating Diffusion
Dynamical systems are commonly used to model mixing in fluid and granular flows. We consider a one-dimensional discontinuous dynamical system model (termed "cutting and shuffling" of a line segment), and we present a comprehensive computational study of finite-time mixing. The properties of the system depend on several parameters in a sensitive way, and the effect of each parameter is examined. Space-time and waterfall plots are introduced to visualize the mixing process with different mixing protocols without diffusion, showing a variety of distinct and complex behaviors in this "simple" dynamical system. To improve the mixing efficiency and avoid pathological cases, we incorporate diffusion into this model dynamical system. We show that diffusion can be quite effective at homogenizing a "mixture." To make this effect clear, we compare cases without diffusion to those with "small" diffusivity and "large" diffusivity. Illustrative examples also show how to adapt mixing metrics from the literature, namely the number of cutting interfaces and a mixing norm, to quantify the degree of mixing in our cutting and shuffling system. To study the evolution of mixing through a large set of possible cutting and shuffling parameters, we introduce fit functions for the number of cutting interfaces and the mixing norm. These fits allow us to determined time constants of mixing for each different system considered, thereby quantifying the "speed" of mixing. Systems with various different permutations (shuffling protocols) are considered, then average properties can be computed, which hold true (on average) for all allowed permutations. The relationship between the fit parameters and the system parameters is also investigated through scatter plots in the fit parameter space. Next, universal mixing behaviors are identified by specifically introducing a critical half-mixing time, which must be found computationally. Using the latter, a rescaling of different dynamical regimes (decay curves of the mixing norm) fall onto a universal profile valid across all parameters of the cutting and shuffling dynamical system. Then, a prediction for this critical half-mixing time is made on the basis of the evolution of the number of subsegments of continuous color (unmixed subsegments). This prediction, which is called a stopping time in the finite Markov chain literature, must also be found numerically. The latter compares well with the previously computed half-mixing time, which provides an approach to determine when a system has become uniform. Finally, we examine the dependence of the half-mixing times on the characteristic Péclet number of the system (an inverse dimensionless diffusivity), and we show that as the Péclet number becomes large, the system transitions more sharply from an unmixed initial state to a mixed final state. This phenomenon, which is know as a "cut-off" in the finite Markov chain literature thus appears to be well substantiated by our numerical investigation of cutting and shuffling a line segment in the presence of diffusion.^
Ivan C. Christov, Purdue University.
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