Optimal mixing in recirculation zones

Abstract

Coarse-scale mixing in a recirculation zone is described with a simple vortex model. Time-dependent forcing is employed to change the vortex motion and mixing properties. An optimal mixing problem is defined in which the flux across the recirculation region shall be maximized under the side-constraints of bounded vortex motion and bounded actuation. Concepts of control theory and chaotic advection are used to achieve this goal. In particular, controllability is proven with a transformation into flat coordinates. Thus, a feedforward law for the optimal trajectory and a feedback law for its stabilization are derived. Observability of the vortex motion is indicated by a dynamic observer. Mixing in the optimized flow is studied using Poincaré maps. The low-frequency modulations to vortex motion are shown to substantially increase mixing in the average. Generalizations of the mathematical framework for mixing optimization are suggested for a larger class of models and flows.

Quantifying Fluid Mixing with the Shannon Entropy

“…[15] In spite of the large amount of work on this topic, there is no consensus on a single measure of mixing that can be used for different processes nor is there a single mixing index that can be implemented easily for different scenarios varying from numerical simulations to laboratory experiments to industrial processes. [16][17][18][19] In this paper we present an approach for measuring the degree of mixing of several fluids that is independent of the physical process responsible for mixing and that relies on the information entropy. This approach is particularly suited for polymerprocessing operations where process control and enhanced Summary: We introduce a methodology to quantify the quality of mixing in various systems, including polymeric ones, by adapting the Shannon information entropy.…”

Controllability for a class of area-preserving twist maps

“…Besides the intrinsic beauty of the subject, this is due to a number of exciting applications such as satellite control [2], quantum control [3,4], and control of mixing [5,6].…”

Approach for maximizing chaotic mixing in microfluidic devices

“…Nevertheless, these often provide little insight into the best mixing protocols that enhance mixing, resulting in very few available investigations on mixing optimization. 12,13,20,22 Recent theoretical work, 22,23 which utilizes lobe dynamics and Melnikov techniques 24,25 to precisely quantify flux across flow separatrices, provides a method to do so. This is particularly relevant to crosschannel micromixers, 7-10 a schematic illustration of which is presented in Fig.…”

“…1-6 The inevitable low Reynolds number limit in such devices means that turbulent mixing is suppressed. Diffusion by itself is not sufficiently effective in obtaining well-mixed solutions in some applications, leading to an interest in deterministic chaos as a mixing mechanism ͓see also the entire issue of [7][8][9][10][11][12][13][14][15][16] Philos. Trans.…”