Topology, Braids, and Mixing in Fluids

Abstract

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, realisable design for a mixing device, the silver mixer, based on these principles.

15 pages, 27 figures. Royal Society style (included). Final version with extra and updated references. A few typos corrected in proofs

References in corpus (4)

• Topological Mixing with Ghost Rods
• Measuring Topological Chaos
• Topological Entropy of Braids on the Torus
• Topological Chaos in Spatially Periodic Mixers

Cited by in corpus (27)

• Frontiers of chaotic advection
• Using multiscale norms to quantify mixing and transport
• Optimal stirring strategies for passive scalar mixing
• Detecting coherent structures using braids
• Braids of entangled particle trajectories
• Finite-Time Braiding Exponents
• Topological optimisation of rod-stirring devices
• Optimizing the Source Distribution in Fluid Mixing
• Topology of Chaotic Mixing Patterns
• Topological Entropy of Braids on the Torus
• The Burau estimate for the entropy of a braid
• Braidlab: A Software Package for Braids and Loops
• Ensemble-based Topological Entropy Calculation (E-tec)
• Trajectory entanglement in dense granular materials
• Lyapunov exponents for the random product of two shears
• On the minimum dilatation of braids on punctured discs
• Nonuniform mixing
• Estimating the rate of field line braiding in the solar corona by photospheric flows
• Using braids to quantify interface growth and coherence in a rotor-oscillator flow
• Topological Chaos in a Three-Dimensional Spherical Fluid Vortex
• Estimating topological entropy from the motion of stirring rods
• The entropy efficiency of point-push mapping classes on the punctured disk
• Topological entropy and secondary folding
• Reynolds number effects on mixing due to topological chaos
• Optimal perturbations for nonlinear systems using graph-based optimal transport
• Is Chaotic Advection Inherent to Heterogeneous Darcy Flow?
• Braided mixing in confined chiral active matter