Topological fluid mechanics of stirring

Abstract

A new approach to regular and chaotic fluid advection is presented that utilizes the Thurston–Nielsen classification theorem. The prototypical two-dimensional problem of stirring by a finite number of stirrers confined to a disk of fluid is considered. The theory shows that for particular ‘stirring protocols’ a significant increase in complexity of the stirred motion – known as topological chaos – occurs when three or more stirrers are present and are moved about in certain ways. In this sense prior studies of chaotic advection with at most two stirrers, that were, furthermore, usually fixed in place and simply rotated about their axes, have been ‘too simple’. We set out the basic theory without proofs and demonstrate the applicability of several topological concepts to fluid stirring. A key role is played by the representation of a given stirring protocol as a braid in a (2+1)-dimensional space–time made up of the flow plane and a time axis perpendicular to it. A simple experiment in which a viscous liquid is stirred by three stirrers has been conducted and is used to illustrate the theory.

Topological Chaos and Periodic Braiding of Almost-Cyclic Sets

“…Each strand returns to its initial position every Át ¼ 3 f ¼ b , which gives one period of the braid. The topology of this braid is identical to that in [7,13].…”

“…TN is the maximum line-stretching exponent over all possible initial material lines. The size of the relevant subdomain is not predicted by the TNCT, but experimental results indicate that this region is typically on the scale of the orbit motions [7].…”

“…One recent approach to analyzing chaos in a fluid system is the identification of ''topological chaos,'' or chaos due to the topology of internal boundary motions such as moving rods [7,8]. This approach is based on application of the Thurston-Nielsen classification theorem [9] (TNCT) to certain (2 þ 1)-dimensional flows: either two-dimensional, time-dependent flows or three-dimensional flows that, because of symmetry, can be cast as surface homeomorphisms (i.e., two-dimensional mappings) [10].…”

“…1(d) is of pseudo-Anosov (PA) type. That is, the underlying flow map is topologically equivalent (i.e., isotopic) to an ideal PA map that, except for a finite number of singularities, stretches everywhere in the unstable direction by a factor TN > 1 and contracts everywhere in the stable direction by a factor 1= TN [7,10]. In the ideal case, material lines throughout the domain grow as p TN , where p is the (integer) period of the braid.…”

“…In [7], the fluid motion is generated by causing solid rods to move along a PA braid. Here, the periodic orbits of the braid are driven by the flow.…”