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Quantifying Mixing using Magnetic Resonance Imaging

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Published on: January 25, 2012

Topology of chaotic mixing patterns.

Jean-Luc Thiffeault¹, Matthew D Finn, Emmanuelle Gouillart

¹Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706, USA. jeanluc@mailaps.org

Chaos (Woodbury, N.Y.)

|December 3, 2008

Summary

This study explores chaotic mixing using a stirring device with rod motions. Topological analysis reveals how unmixed material is injected into the central region, aiding efficient mixing.

Area of Science:

Fluid dynamics
Topology
Chaos theory

Background:

Chaotic mixing is crucial for efficient fluid processing.
Understanding the topological nature of fluid motion aids in designing effective mixing devices.
Periodic rod motions can induce complex fluid mappings.

Purpose of the Study:

To analyze the fluid dynamics induced by a periodic rod motion stirring device.
To investigate the topological properties of the induced fluid mapping.
To understand the mechanism of unmixed material injection into a central mixing region.

Main Methods:

Modeling the stirring device as a homeomorphism of a punctured surface.
Utilizing topological considerations to analyze material line stretching and chaotic mixing.

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Applying a topological index formula to predict unstable foliations.

Main Results:

The periodic motion of rods creates a fluid domain mapping, acting as a homeomorphism.
Topologically complex rod motion ensures material line stretching, essential for chaotic mixing.
Injection cusps are identified as the sites for unmixed material injection into the central region.
A topological index formula successfully predicts possible unstable foliations for a given number of rods.

Conclusions:

The stirring device effectively induces chaotic mixing through topological fluid mapping.
Topological analysis provides a framework for understanding and predicting mixing efficiency.
The study offers insights into the design of advanced fluid mixing systems.