Convection-enhanced diffusion for random flows |
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Authors: | Albert Fannjiang George Papanicolaou |
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Institution: | (1) Department of Mathematics, University of California at Davis, 95616 Davis, California;(2) Department of Mathematics, Stanford University, 94305 Stanford, California |
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Abstract: | We analyze the effective diffusivity of a passive scalar in a two-dimensional, steady, incompressible random flow that has
mean zero and a stationary stream function. We show that in the limit of small diffusivity or large Peclet number, with convection
dominating, there is substantial enhancement of the effective diffusivity. Our analysis is based on some new variational principles
for convection diffusion problems and on some facts from continuum percolation theory, some of which are widely believed to
be correct but have not been proved yet. We show in detail how the variational principles convert information about the geometry
of the level lines of the random stream function into properties of the effective diffusivity and substantiate the result
of Isichenko and Kalda that the effective diffusivity behaves likeɛ
3/13 when the molecular diffusivityɛ is small, assuming some percolation-theoretic facts. We also analyze the effective diffusivity for a special class of convective
flows, random cellular flows, where the facts from percolation theory are well established and their use in the variational
principles is more direct than for general random flows. |
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Keywords: | Diffusion convection random media percolation |
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