Topology of Chaotic Mixing Patterns
Résumé
A stirring device consisting of a periodic motion of rods induces a mapping of the fluid domain to itself, which can be regarded as a homeomorphism of a punctured surface. Having the rods undergo a topologically-complex motion guarantees a minimal amount of stretching of material lines, which is important for chaotic mixing. We use topological considerations to describe the nature of the injection of unmixed material into a central mixing region, which takes place at injection cusps. The number and position of injection cusps is particularly important for open flows, such as flows in channels, where the cusps determine the shape of the downstream mixing pattern. A topological index formula allow us to predict the possible types of unstable foliations that can arise for a fixed number of rods.