Numerical Evidence for Cutoffs in Chaotic Microfluidic Mixing

Chaotic mixing strategies produce high mixing rates in microfluidic channels and other applications. In prior numerical and experimental work the variance of a tracer field in a chaotic mixer has been observed to decay rapidly after an initial slower transient. We relate this to the cutoff phenomenon observed in finite Markov chains and provide numerical evidence to suggest that chaotic mixing indeed exhibits cutoff. We provide results for a herringbone passive microfluidic mixer and the Standard Map. INTRODUCTION The question of how chaotic advection mixes a passive scalar function has attracted much research effort in recent years [1]. The main issues in this field are: how to measure the thoroughness of the mixing, how the mixing process changes qualitatively and quantitatively when the diffusion is close to zero, and how to enhance the overall mixing process by designing the map which produces chaotic advection. Unfortunately, we have only partial understanding for most of these topics. In spite of the fact that the detailed mechanism of mixing is unclear, nontrivial mixing processes have been observed in experiments [2] and can be simulated by large-scale computations [3]. A widely observed phenomenon in the chaotic mixing process when small diffusion exists is the two or three-stage transition [4–6]. The map does not mix the scalar function with a constant rate in general. When the variance of the scalar function is measured during the mixing process, one can in general observe a relatively flat decay initially, followed by a super-exponential change, and then finally it tends to an exponential decay. We are interested in when these transitions happen, why they happen, and how to predict the slope of the exponential region. A good review and physical interpretation can be found in [7]. Thiffeault and Childress [4] study these properties for a modified Arnold’s cat map. Analytical formulas are given to predict the transitions as well as the slopes. Because the linear part of this map has an eigenvalue 2.618, which stretches very fast, and the chaotic part is relatively small, the three phases are separated clearly. The same analytical procedure cannot be applied to, for example, the Standard Map, although the only difference between the Standard Map and the modified Arnold’s cat map is in the linear part. As for the exponential decay part, there is still debate about whether the decay rate goes to zero in the zero diffusivity limit or whether it tends to a constant independent of the diffusion [3,7]. Theoretical analysis shows both of these possibilities can occur for different chaotic flows [8]. Difficulties typically arise in studying the above problems numerically, because the small diffusion usually means that fine grids are required in the solution of the advection-diffusion equation or the simulation of the map. Some studies and numerical results conclude that a proportional relation exists between the stationary decay rate and the diffusion [9]. However, this is only true for certain diffusion ranges. Our goal in this paper is to relate the chaotic mixing process to the well-known cutoff phenomenon in finite Markov Chain studies (see [10] and references therein). We begin with a numerical simulation of a chaotic mixing channel, measuring mixing of two colored liquids by the color variance of channel crosssections. The simulation shows that when we increase the Péclet number, the mixing trajectory presents a cutoff. The underlying physical mechanism is then explained using advection of a sinusoidal function under the Baker’s Map. To support the chaotic mixing channel example, a very high resolution numerical simulation of the Standard Map is then presented to show that in the near-zero diffusion limit it does present a cutoff. BACKGROUND The measure space and operators We work on the probability space (X ,A ,μ) for X a subset of Rn. We take S : X → X to be a transformation (or map) that is non-singular and measurable. We choose μ to be the Borel measure. In the measure space (X ,A ,μ) we define the following operators. Definition 1. (Perron-Frobenius operator) The PerronFrobenius operator P : L1(X) → L1(X) associated with S satisfies