Diffusion in arrays of obstacles: Beyond homogenization: Diffusion in arrays of obstacles

Yahya Farah, Daniel Loghin, Alexandra Tzella*, Jacques Vanneste

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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. Homogenization 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 confirms the validity of the latter.

Original languageEnglish
Article number20200207
JournalProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2244
Publication statusPublished - Dec 2020

Bibliographical note

Funding Information:
Funding. Y.F.’s research was supported by a PhD scholarship from the EPSRC. J.V.’s research was supported by EPSRC Programme Grant EP/R045046/1: Probing Multiscale Complex Multiphase Flows with Positrons for Engineering and Biomedical Applications (PI: Prof. M. Barigou, University of Birmingham). Acknowledgements. A. Tzella and D. Loghin acknowledge A. Patel for his contribution to a related MSci project.

Publisher Copyright:
© 2020 The Author(s).


  • composite materials
  • diffusion
  • homogenization
  • large deviations
  • porous media

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)


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