Mathematical modelling of liquid transport in porous materials at low levels of saturation

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

Capillary transport in paper porous materials at low saturation levels: normal, fast or superfast?

The problem of capillary transport in fibrous porous materials at low levels of liquid saturation has been addressed. It has been demonstrated, that the process of liquid spreading in this type of porous materials at low saturation can be described macroscopically by a similar super-fast, non-linear diffusion model as that, which had been previously identified in experiments and simulations in particulate porous media. The macroscopic diffusion model has been underpinned by simulations using a microscopic network model. The theoretical results have been qualitatively compared with available experimental observations within the witness card technique using persistent liquids. The long-term evolution of the wetting spots was found to be truly universal and fully in line with the mathematical model developed. The result has important repercussions on the witness card technique used in field measurements of dissemination of various low volatile agents in imposing severe restrictions on collecting and measurement times

Surface permeability of porous media particles and capillary transport

We have established previously, in a lead-in study, that the spreading of liquids in particulate porous media at low saturation levels, characteristically less than 10% of the void space, has very distinctive features in comparison to that at higher saturation levels. In particular, we have found that the dispersion process can be accurately described by a special class of partial differential equations, the super-fast non-linear diffusion equation. The results of mathematical modelling have demonstrated very good agreement with experimental observations. However, any enhancement of the accuracy and predictive power of the model, keeping in mind practical applications, requires the knowledge of the effective surface permeability of the constituent particles, which defines the global, macroscopic permeability of the particulate media. In the paper, we demonstrate how this quantity can be determined through the solution of the Laplace-Beltrami Dirichlet problem, we study this using the well-developed surface finite element method

Surface permeability of particulate porous media

The dispersion process in particulate porous media at low saturation levels takes place over the surface elements of constituent particles and, as we have found previously by comparison with experiments, can be accurately described by super-fast non-linear diffusion partial differential equations. To enhance the predictive power of the mathematical model in practical applications, one requires the knowledge of the effective surface permeability of the particle-in-contact ensemble, which can be directly related with the macroscopic permeability of the particulate media. We have shown previously that permeability of a single particulate element can be accurately determined through the solution of the Laplace-Beltrami Dirichlet boundary-value problem. Here, we demonstrate how that methodology can be applied to study permeability of a randomly packed ensemble of interconnected particles. Using surface finite element techniques we examine numerical solutions to the Laplace-Beltrami problem set in the multiply-connected domains of interconnected particles. We are able to directly estimate tortuosity effects of the surface flows in the particle ensemble setting

Weighted (<i>E</i>_λ, <i>q</i>)(<i>C</i>_λ, 1) Statistical Convergence and Some Results Related to This Type of Convergence

In this paper, we defined weighted (Eλ,q)(Cλ,1) statistical convergence. We also proved some properties of this type of statistical convergence by applying (Eλ,q)(Cλ,1) summability method. Moreover, we used (Eλ,q)(Cλ,1) summability theorem to prove Korovkin’s type approximation theorem for functions on general and symmetric intervals. We also investigated some of the results of the rate of weighted (Eλ,q)(Cλ,1) statistical convergence and studied some sequences spaces defined by Orlicz functions