Topological fluid mechanics of stirring

A new approach to regular and chaotic fluid advection is presented that uti lizes the Thurston-Nielsen classification theorem. The prototypical two-dim ensional problem of stirring by a finite number of stirrers confined to a d isk of fluid is considered. The theory shows that for particular 'stirring protocols' a significant increase in complexity of the stirred motion - kno wn as topological chaos - occurs when three or more stirrers are present an d are moved about in certain ways. In this sense prior studies of chaotic a dvection with at most two stirrers, that were, furthermore, usually fixed i n place and simply rotated about their axes, have been 'too simple'. We set out the basic theory without proofs and demonstrate the applicability of s everal 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)-dimension al 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 stirrer s has been conducted and is used to illustrate the theory.