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Does a fast mixer really exist?
1Department of Mathematics, University of Central Arkansas, 201 Donaghey Avenue, Conway, Arkansas 72034, USA. liuweijiu@hotmail.com
Summary
Researchers investigated how scalar variance decay rates behave as diffusivity approaches zero in fluid flows. Simulations and theory indicate that the decay rate tends to zero, suggesting fast mixers are difficult to find.
Area of Science:
- Fluid dynamics
- Statistical mechanics
- Dynamo theory
Background:
- Investigating the decay rate of variance for passive and diffusive scalars in flow fields.
- Exploring the concept of 'fast mixers' in fluid dynamics, analogous to fast dynamos in dynamo theory.
- Examining the limit behavior as scalar diffusivity approaches zero.
Purpose of the Study:
- To determine if the decay rate of scalar variance remains non-zero as diffusivity approaches zero.
- To analyze the existence and properties of 'fast mixer' flows.
- To theoretically and numerically investigate scalar transport in various flow fields.
Main Methods:
- Numerical simulations using established flow maps (lattice, 1D baker's, sinusoidal shear).
- Theoretical analysis for closed flows in bounded domains.
- Analysis of effective diffusivity for open flows in the whole space.
Main Results:
- Simulations consistently showed the decay rate tending to zero as diffusivity approached zero.
- Theoretical proof for closed flows under plausible conditions confirmed this trend.
- For open flows, the effective diffusivity matrix was shown to approach zero in the limit.
Conclusions:
- The study suggests that flows previously hypothesized as 'fast mixers' do not exhibit this property under the examined conditions.
- Finding a true 'fast mixer' flow appears challenging based on current theoretical and numerical evidence.
- The decay rate of scalar variance generally tends to zero as diffusivity diminishes, irrespective of flow type.