# Analysis of Enhanced Diffusion in Taylor Dispersion via a Model Problem

## Abstract

We consider a simple model of the evolution of the concentration of a tracer, subject to a background shear flow by a fluid with viscosity *ν* ≪ 1 in an infinite channel. Taylor observed in the 1950s that, in such a setting, the tracer diffuses at a rate proportional to 1∕*ν*, rather than the expected rate proportional to *ν*. We provide a mathematical explanation for this enhanced diffusion using a combination of Fourier analysis and center manifold theory. More precisely, we show that, while the high modes of the concentration decay exponentially, the low modes decay algebraically, but at an enhanced rate. Moreover, the behavior of the low modes is governed by finite-dimensional dynamics on an appropriate center manifold, which corresponds exactly to diffusion by a fluid with viscosity proportional to 1∕*ν*.

## Notes

### Acknowledgements

The work of OC and CEW was supported in part by the NSF through grant DMS-1311553. The work of MB was supported in part by a Sloan Fellowship and NSF grant DMS-1411460. MB and CEW thank Tasso Kaper and Edgar Knobloch for pointing out a possible connection between their prior work in [2] and the phenomenon of Taylor dispersion, and we all gratefully acknowledge the many insightful and extremely helpful comments of the anonymous referee.

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