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Topology, braids and mixing in fluids
Jean-Luc Thiffeault1, Matthew D Finn
1Department of Mathematics, Imperial College London, London SW7 2AZ, UK. jeanluc@imperial.ac.uk
Summary
Fluid mixing relies on stirring rods that create topological obstacles, leading to stretching and folding. Analyzing rod trajectories as braids helps design efficient fluid mixing devices, like the novel silver mixer.
Area of Science:
- Fluid dynamics
- Topology
- Chemical Engineering
Background:
- Stirring fluids with moving rods is crucial for achieving homogeneity in various applications.
- These rods act as topological obstacles, inducing fluid element stretching and folding, which are precursors to effective mixing.
- The study of fluid mixing benefits from a topological perspective, viewing rod movements as braids.
Purpose of the Study:
- To review the topological viewpoint of fluid mixing.
- To discuss the application of braid properties in diagnosing mixing efficiency.
- To introduce a new, realizable design for an efficient mixing device based on topological principles.
Main Methods:
- Analysis of fluid element stretching and folding caused by moving rods.
- Representation of rod trajectories in a space-time diagram as braids.
- Utilizing braid properties to understand and enhance fluid mixing.
Main Results:
- The stretching and folding of fluid elements, observed as filaments and striations, are direct consequences of topological obstacles.
- The complexity of the flow is minimally constrained by the properties of the braid formed by rod trajectories.
- Braids offer a method to diagnose mixing efficiency and guide the design of mixing devices.
Conclusions:
- A topological approach, utilizing braid theory, provides valuable insights into fluid mixing processes.
- The properties of braids can be leveraged to construct more efficient fluid mixing devices.
- A novel, realizable mixing device, the 'silver mixer,' has been designed based on these topological mixing principles.
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