Transport in periodic porous media: beyond homogenisation

Farah, Yahya (2024). Transport in periodic porous media: beyond homogenisation. University of Birmingham. Ph.D.

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Abstract

The dispersion of scalars (or heat) inside a fluid flowing through a porous medium is often examined using the theory of homogenisation. 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 of the scalar is approximately Gaussian. This thesis improves on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the scalar concentration through a rate function obtained by solving a family of eigenvalue problems. We demonstrate this on two distinct examples of idealised porous media. The first example is a medium composed of a periodic array of impermeable cylindrical obstacles. We focus on the classical problem of diffusion and examine the dilute and dense limits, when the obstacles occupy a small and large area fraction, respectively. We derive asymptotic approximations for the rate function that explain the validity of the Gaussian behaviour in the dilute limit and capture the non-Gaussian behaviour in the dense limit. 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 in the dense limit. Comparison between numerical results and asymptotic predictions confirm the validity of the latter. The second example is a periodic network composed of one-dimensional edges along which fluid flows with uniform velocity. We focus on networks generated from Bravais (triangular and square) and non-Bravais (hexagonal) lattices. We derive a set of transcendental equations from where the rate function can be extracted, yielding the effective diffusivity tensor that governs the Gaussian approximation as a byproduct. The dependence of dispersion on the underlying geometry and topology is determined by examining a set of asymptotic approximations for the effective diffusivity tensor and the rate function in a variety of physical regimes.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):