By Andrew H Wallace
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Extra resources for An introduction to algebraic topology
6 Egg beater flows Franjione & Ottino (1992) developed a model that encapsulates the essential kinematic mechanisms for ‘good mixing’ in a large class of mixers. The model was arrived at as a result of the accumulation of a great deal of experience analyzing a variety of diverse mixing situations from the dynamical systems point of view. Interestingly, and unrecognized at the time, the model is precisely a linked twist map on the torus. 9 (a) Schematic of the egg beater. In (b) a blade pushes a material line, in (c) a second blade folds the line.
17. Of course, there is a slight complication with this picture. In this analogy the vertical sides of a square are being collapsed to a point. In other words, if a fluid particle goes down a sink, what ‘direction’ does it come out of the source? g. ). Also, the standard ‘twist condition’ on the ‘centreline’ connecting source-sink pairs breaks down. Nevertheless, the LTM framework provides a framework, and a variety of tools, for rigorous mixing studies of this system. 6 Mixing at the microscale Mixing at the microscale is an increasingly important subject that can be analysed in detail by the methods developed in this book.
However, forming a blob in granular matter is hard and very quickly the blob becomes broken and connectivity of the ‘dyed’ structure, as opposed to the companion fluid case, is lost. Segregation experiments in granular matter, on the other hand, are easy and 26 1 Mixing: physical issues can be repeated multiple times. As indicated earlier a distinguishing feature of flowing granular matter is its tendency to segregate; mixtures of particles with varying size (S-systems) or varying density (D-systems) subject to flow often segregate leading to what on first viewing appear to be baffling results.