Topological chaos relies on the periodic motion of obstacles in a two-dimensional flow in order to form nontrivial braids. This motion generates exponential stretching of material lines, and hence efficient mixing. Boyland et al. [P. L. Boyland, H. Aref, and M. A. Stremler, J. Fluid Mech. 403, 277 (2000)] have studied a specific periodic motion of rods that exhibits topological chaos in a viscous fluid. We show that it is possible to extend their work to cases where the motion of the stirring rods is topologically trivial by considering the dynamics of special periodic points that we call ghost rods, because they play a similar role to stirring rods. The ghost rods framework provides a new technique for quantifying chaos and gives insight into the mechanisms that produce chaos and mixing. Numerical simulations for Stokes flow support our results.
Stirring of fluid with moving rods is necessary in many practical applications to achieve homogeneity. These rods are topological obstacles that force stretching of fluid elements. The resulting stretching and folding is commonly observed as filaments and striations, and is a precursor to mixing. In a space-time diagram, the trajectories of the rods form a braid, and the properties of this braid impose a minimal complexity in the flow. We review the topological viewpoint of fluid mixing, and discuss how braids can be used to diagnose mixing and construct efficient mixing devices. We introduce a new, realizable design for a mixing device, the silver mixer, based on these principles.
Topological chaos may be used to generate highly effective laminar mixing in a simple batch stirring device. Boyland, Aref & Stremler (2000) have computed a material stretch rate that holds in a chaotic flow, provided it has appropriate topological properties, irrespective of the details of the flow. Their theoretical approach, while widely applicable, cannot predict the size of the region in which this stretch rate is achieved. Here, we present numerical simulations to support the observation of Boyland et al. that the region of high stretch is comparable with that through which the stirring elements move during operation of the device. We describe a fast technique for computing the velocity field for either inviscid, irrotational or highly viscous flow, which enables accurate numerical simulation of dye advection. We calculate material stretch rates, and find close agreement with those of Boyland et al., irrespective of whether the fluid is modelled as inviscid or viscous, even though there are significant differences between the flow fields generated in the two cases. IntroductionStatic and dynamic mixing devices are important in many industries, e. ) have demonstrated, in an unusual blend of ad hoc experimentation and abstract mathematics, that flows with the topology of certain braids achieve a material stretch rate which can be determined quantitatively, given only the topology of the flow. However, a key feature not predicted by their theoretical considerations is the size of the domain in which this stretch rate is attained. Indeed, according to the theory, this domain may have measure zero, and if this were the case then the theory would have little practical impact. Here we provide numerical results that support the observations of Boyland et al., that the chaotic region is in fact commensurate with the region of fluid through which the stirring elements move during operation of the device. We should make clear at the outset that we use the terminology 'topological chaos' in the same sense as Boyland et al. (2000), to
An attractive method for valorization of glycerol is the catalytic transformation to lactic acid. By overcoming the solubility challenge associated with known homogeneous catalysts for this reaction, we show that thermally robust Ir(I), Ir(III), and Ru(II) N-heterocyclic carbene (NHC) complexes with sulfonate-functionalized wingtips are highly prolific for this process, requiring no cosolvents other than aqueous base. The activity of the catalysts is compared under both conventional heating and microwave conditions. The most active catalyst reaches a TOF of 45 592 h −1 (microwave) and 3477 h −1 (conventional) with 1 equiv of KOH, and proceeds at a constant rate for at least 8 h. Although higher activity is observed with KOH, the catalysts are also highly active with the weaker base, K 2 CO 3 (13 000 h −1 and concurrent formation of formate). The protocol can be modified to achieve quantitative conversion of glycerol in only 3 h. The high activity of these catalysts compared to nonsulfonated analogs is attributed to the stabilization the lactate product in aqueous media. The most active catalyst retains equal activity for crude glycerol. A mechanism is proposed for the most active catalyst precursor involving O−H oxidative addition of glycerol.
There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly stretch a material such as bread dough or taffy. The goal in these applications is to stretch either material lines (in a fluid) or the material itself (for dough or taffy) as rapidly as possible. The growth rate of material lines is conveniently given by the topological entropy of the rod motion. We discuss the problem of optimising such rod devices from a topological viewpoint. We express rod motions in terms of generators of the braid group, and assign a cost based on the minimum number of generators needed to write the braid. We show that for one cost functionthe topological entropy per generator-the optimal growth rate is the logarithm of the golden ratio. For a more realistic cost function, involving the topological entropy per operation where rods are allowed to move together, the optimal growth rate is the logarithm of the silver ratio, 1 + √ 2. We show how to construct devices that realise this optimal growth, which we call silver mixers.
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