Abstract/Summary

Persistent (low volatility) liquids can disseminate significantly in porous substrates and wet large volumes before they are removed by evaporation or chemical processes. The liquid saturation during this kind of dissemination process quickly reaches very low levels, well below 10% of the available void space, and the liquid dispersion enters a special regime of spreading, when the transport predominantly occurs over surface elements of the porous matrix. On the ’macroscopic’ level, this phenomenon can be described by a special super-fast non-linear diffusion model. But, the model requires the knowledge of permeability coefficients defined by ’microscopic’ mechanisms. The focus of this study is on the mathematical problems associated with the ’microscopic’ level, that is on the details of the surface diffusion processes to obtain accurate definitions of the ’macroscopic’ parameters. We consider two kinds of porous structures with representative properties, paper-like and particulate porous materials, and as a result, two different model approaches, a network model and a surface diffusion model based on the Laplace-Beltrami operator and on the associated LaplaceBeltrami boundary value problems. We demonstrate their feasibility by applying numerical methods, specifically, surface finite elements techniques. We will show, in the Thesis, that the network model is capable of accurately reproducing macroscopic descriptions of the fibrous material, while at the same time providing necessary permeability coefficients of the porous network with minimal assumptions. In the case of particulate porous media, we will demonstrate that, solutions to the Laplace-Beltrami boundary value problem can be used to obtain surface permeability of both single porous matrix elements and their interconnected compositions. We will also demonstrate, for the first time, how effects of tortuosity of the surface flow can be easily obtained while analysing solutions of the Laplace-Beltrami boundary value problem set in the multiply-connected domains formed by mutually coupled particles. Overall, results of this study will improve our understanding of microscopic dispersion processes central to applications of macroscopic descriptions formulated at low saturation levels. Numerical studies of the Laplace-Beltrami boundary value problem using the surface finite element method are interesting on their own, since they demonstrate that similar convergence rates (using relatively standard surface element settings) can be achieved in the domains with smooth boundaries to those regularly observed in the problems without domain boundaries. Therefore, due to the fundamental advances achieved in the study, the macroscopic descriptions used in practice at low saturation levels obtained rigorous foundation and practical recipes, which can be directly used in applications.