Diffusion in arrays of obstacles: beyond homogenisation

Yahya Farah, Daniel Loghin, Alexandra Tzella, Jacques Vanneste

Research output: Contribution to journalArticlepeer-review

Abstract / Description of output

We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenisation theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirm the validity of the latter.
Original languageEnglish
Article number20200072
Number of pages21
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume476
DOIs
Publication statusPublished - 23 Dec 2020

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