Transport Processes in Spatially Disordered Systems

UoM administered thesis: Phd

  • Authors:
  • Matthew Russell

Abstract

Transport phenomena in nature occur in spatially complex and heterogeneous systems, such as the lung and the placenta. Motivated by such examples, we seek to characterise the effects of and interaction between multiscale spatial structure and disorder in models of transport and flow. We first investigate an individual-based transport model which incorporates two sources of noise and transport processes occurring over disparate lengthscales. Particles can hop stochastically between sites on a lattice in one spatial dimension and are taken up via first-order kinetics at discrete sinks with random strengths, sparsely but periodically located along the domain. Simulations indicate that disordered sink strengths induce long-range correlations in the particle concentration which are absent when sinks are of fixed strength. Furthermore, mean concentrations are elevated by disorder. Exploiting the separation of lengthscales and properties of the stochastic hopping, we use a continuum limit to derive approximate, homogenized expressions describing the correlations and elevated mean. These involve non-local combinations of the leading-order concentration profile and the Green's function associated with the corrections. We show that the correction to the mean concentration is always non-negative and so the leading-order classical homogenization approximation underestimates the mean. We finally classify the regions of parameter space according to the dominant physical processes in each region and put bounds on the validity of the homogenization approach by analysing the sizes of fluctuations due to disorder. We next study a related model of solute transport in one dimension where sinks of equal strength are distributed randomly along a line. Extending the Green's function approach, we investigate the interactions between the discrete nature of the sinks and their disordered locations. We find that classical two-scale expansion-based homogenization fails to accurately predict higher-order corrections and develop refined predictions. These methods extend to handling spatially disordered sinks, and we demonstrate that our predictions of the spatial correlations and corrections to leading-order means agree well with simulations in large areas of parameter space and for both weak and strong disorder. The strength of the spatial disorder can affect whether the discrete-to-continuous or disorder effects are the most important. Finally, we study the flow through a two-dimensional disordered porous medium. We perform direct numerical simulations of a low Reynolds number flow past a series of obstacles. Applying spatial perturbations to individual obstacles in an otherwise periodic arrangement, and comparing with a completely periodic array, we investigate the long-range effects in the velocity and pressure fields. This is effective a numerical approximation of a series of Green's functions, inspired by the theoretical framework used in our previous models. We then generate large ensembles of weakly and strongly disordered porous media and compare the statistics of the flow rate with predictions made by Darcy's law, typically applicable to large, periodic arrays. Depending on the type of spatial disorder, mean flow rates can be either elevated or diminished compared with a periodic array and Darcy's law. Furthermore, we analyse the decay in the fluctuations around the mean flow rate as the size of the array is increased.

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Original languageEnglish
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Award date1 Aug 2018