Diffusion in arrays of obstacles: beyond homogenisation
Duration: 54 mins 8 secs
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Description: |
Alexandra Tzella (University of Birmingham)
9 March 2022 – 13:30 to 14:30 |
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Created: | 2022-03-14 13:52 |
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Collection: | Mathematical aspects of turbulence: where do we stand? |
Publisher: | Isaac Newton Institute |
Copyright: | Alexandra Tzella |
Language: | eng (English) |
Abstract: | 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 con- centration 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. |
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