Abstract.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 nonlinear 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 LaplaceBeltrami Dirichlet problem, we study this using the well-developed surface finite-element method.

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 rigorously estimate tortuosity effects of the surface flows in a particle ensemble setting.

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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.