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We study numerically the tortuosity-porosity relation in a microscopic model of a porous medium arranged as a collection of freely overlapping squares. It is demonstrated that the finite-size, slow relaxation and discretization errors, which were ignored in previous studies, may cause significant underestimation of tortuosity. The simple tortuosity calculation method proposed here eliminates the need for using complicated, weighted averages. The numerical results presented here are in good agreement with an empirical relation between tortuosity (T) and porosity (varphi) given by T-1 proportional, variantlnvarphi , that was found by others experimentally in granule packings and sediments. This relation can be also written as T-1 proportional, variantRSvarphi with R and S denoting the hydraulic radius of granules and the specific surface area, respectively. Applicability of these relations appears to be restricted to porous systems of randomly distributed obstacles of equal shape and size.

Tortuosity (T ) is a parameter describing an average elongation of fluid streamlines in a porous medium as compared to free flow. In this paper several methods of calculating this quantity from lengths of individual streamlines are compared and their weak and strong features are discussed. An alternative method is proposed, which enables one to calculate T directly from the fluid velocity field, without the need of determining streamlines, which greatly simplifies determination of tortuosity in complex geometries, including those found in experiments or 3D computer models. Numerical results obtained with this method suggest that (a) the hydraulic tortuosity of an isotropic fibrous medium takes on the form T = 1 + p √ 1 − ϕ, where ϕ is the porosity and p is a constant and (b) the exponent controlling the divergence of T with the system size at percolation threshold is related to an exponent describing the scaling of the most probable traveling length at bond percolation.

We examine the long time behaviour of A + B → 0 reaction diffusion systems with initially segregated species A and B. All of our analysis is carried out for arbitrary (positive) values of the diffusion constants D A , D B , and initial concentrations a 0 and b 0 of A's and B's. We divide the domain of the partial differential equations describing the problem into several regions in which they can be reduced to simpler, solvable equations, and we merge the solutions. Thus we derive general formulae for the concentration profiles outside the reaction zone, the location of the reaction zone center, and the total reaction rate. An asymptotic condition for the reaction front to be stationary is also derived. The properties of the reaction layer are studied in the mean-field approximation, and we show that not only the scaling exponents, but also the scaling functions are independent of D A , D B , a 0 and b 0 .

We develop a practical method of computing the stationary drift velocity V and the diffusion coefficient D of a particle (or a few particles) in a periodic system with arbitrary transition rates. We solve this problem both in a physically relevant continuous-time approach as well as for models with discrete-time kinetics, which are often used in computer simulations. We show that both approaches yield the same value of the drift, but the difference between the diffusion coefficients obtained in each of them equals 1 2 V 2 . Generalization to spaces of arbitrary dimension and several applications of the method are also presented.