Matthew D. Finn

Topological chaos in inviscid and viscous mixers

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. ,i rrespective 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.

Topology, braids and mixing in fluids

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 in spatially periodic mixers

Topologically chaotic fluid advection is examined in two-dimensional flows with either or both directions spatially periodic. Topological chaos is created by driving flow with moving stirrers whose trajectories are chosen to form various braids. For spatially periodic flows, in addition to the usual stirrer-exchange braiding motions, there are additional topologically nontrivial motions corresponding to stirrers traversing the periodic directions. This leads to a study of the braid group on the cylinder and the torus. Methods for finding topological entropy lower bounds for such flows are examined. These bounds are then compared to numerical stirring simulations of Stokes flow to evaluate their sharpness. The sine flow is also examined from a topological perspective.

Topological Optimization of Rod-Stirring Devices

There are many industrial situations where rods are used to stir a fluid, or where rods repeatedly knead 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 optimizing 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 function—the 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,

Chaotic advection in a braided pipe mixer

1+\sqrt{2}

Two-dimensional Stokes flow driven by elliptical paddles

. We show how to construct devices that realize this optimal growth, which we call silver mixers.

Evanescent coupling between discs: a model for near-integrable tunnelling

Chaotic advection is investigated in a “braided pipe mixer” (BPM). This static mixing device consists of an outer pipe containing several intertwining internal pipes, with fluid pumped down the gap. It has recently been proposed, by analogy with the two-dimensional theory of topological chaos, that the BPM should be an effective mixing device, specifically that (i) good mixing might be achieved with only very thin internal pipes, and (ii) the quality of the mixing should be improved if the internal pipes form a mathematical braid. Our results suggest that neither (i) nor (ii) is the case for the BPM studied.

The Size of Ghost Rods

A fast and accurate numerical technique is developed for solving the biharmonic equation in a multiply connected domain, in two dimensions. We apply the technique to the computation of slow viscous flow (Stokes flow) driven by multiple stirring rods. Previously, the technique has been restricted to stirring rods of circular cross section; we show here how the prior method fails for noncircular rods and how it may be adapted to accommodate general rod cross sections, provided only that for each there exists a conformal mapping to a circle. Corresponding simulations of the flow are described, and their stirring properties and energy requirements are discussed briefly. In particular the method allows an accurate calculation of the flow when flat paddles are used to stir a fluid chaotically.

Topological mixing with ghost rods.

An analysis is provided of the splitting due to tunnelling of the energy levels in two-dimensional double-disk potentials. The formal setting of this problem is similar to that of tunnelling between tori in the near-integrable regime but is free of the difficulties arising from the existence of natural boundaries that are generic to such problems, enabling a systematic investigation of the relevant semiclassical theory. The semiclassical predictions are found to be consistent with exponentially accurate results obtained following reduction to a boundary-value problem. A numerical quantization is also performed and found to agree with the approximate results.

Mixing measures for a two-dimensional chaotic Stokes flow

“Ghost Rods” are periodic structures in a two-dimensional flow that have an effect on material lines that is similar to real stirring rods. An example is a periodic island: material lines exterior to it must wrap around such an island, because determinism forbids them from crossing through it. Hence, islands act as topological obstacles to material lines, just like physical rods, and lower bounds on the rate of stretching of material lines can be deduced from the motion of islands and rods. Here, we show that unstable periodic orbits can also act as ghost rods, as long as material lines can “fold” around the orbit, which requires the orbit to be parabolic. We investigate the factors that determine the effective size of ghost rods, that is, the magnitude of their impact on material lines.