Geometric Control of Asymmetries in Passive Scalars Advected in Rectangular and Elliptical Pipes

Abstract eng:
We explore the role different geometries (amongst rectangular and elliptical domains of arbitrary aspect ratios) play in controlling emerging up-stream/downstream asymmetries in the cross-sectionally averaged distribution of diffusing passive scalars advected by laminar, pressure driven shear flows. We show using a combination of rigorous analysis, asymptotic expansions, and Monte-Carlo simulations, that on short time scales relative to the shortest diffusion times, elliptical domains preserve initial uptream/downstream symmetric distributions, while rectangular ducts break this symmetry. Skinny ducts produce distributions with negative skewness, while fat ducts produce positive skewness for symmetric initial data which is uniformly distributed in the cross-section. There is a special aspect ratio of approximately 2-1 ratio for which symmetry is preserved. In turn, long-time (relative to the longest diffusion timescale) exact analysis shows that all geometries generically break symmetry before ultimately symmetrizing in infinite time.

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