A topological approach to 3D laminar mixing
by
Nathaniel Jewell
The University of Adelaide
Coauthors: Matthew Finn

In recent years, topological concepts have yielded valuable insights into the long-standing problem of laminar fluid mixing. Topologically-complex stirring protocols are typically far superior to topologically-simple protocols, guaranteeing chaotic advection of fluid particles and the associated exponential dilation of material elements. Furthermore, topological approaches to mixer design are typically intuitive and insensitive to precise geometry or fluid properties. However, published results to date have been limited to 2D flows (e.g. batch-stirrers in food or polymer manufacturing) and quasi-3D protocols (e.g. continuous-flow micromixers). Motivated by a simple stretching and folding argument that works well in 2D, we propose a topological approach to fully three-dimensional fluid mixing. A transition matrix can be derived to describe the mapping induced by a 3D "braid" on area elements, and the associated Perron-Frobenius eigenvalue provides a prediction of the large-time asymptotic area growth rate. We show that these theoretical predictions agree very well with numerical data obtained from simulations in a prototype 3D mixing device.

Date received: October 20, 2009