Simulation of concentration of an admixture in the multiphase layer with random spherical inclusions

The diffusion of an admixture substance in a multiphase layer with randomly disposed spherical inclusions was investigated. The solution of the initial contact-boundary value problem is obtained in the form of the integral Neumann series. Computer simulation was performed based on the obtained calculation formula. Main regularities of the distributions of the averaged admixture concentration in the layer depending on the values of the diffusion coefficients, density and volume fractions of inclusions were established. The influence of the number of phases of the porous body on the diffusion processes in a multiphase layer with a uniform distribution of spherical inclusions was determined. The dependence of the increase of the averaged concentration function on the characteristic radii of spherical inclusions was analyzed, in particular, it is shown that the behavior of this function does not depend on the ratios of the reduced diffusion coefficients.

A simple justification of effective models for conducting or fluid media with dilute spherical inclusions

We present a gentle approach to the justification of effective media approximations, for PDE’s set outside the union of n ≫ 1 spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term, made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as n → + ∞ by simple arguments.

Mathematical modeling diffusion of admixture particles in a strip with randomly located spherical inclusions of different materials with commensurable volume fractions of phases

The process of diffusion of admixture particles in a multiphase randomly nonhomogeneous body with spherical inclusions of different materials with commensurable volume fractions of phases is investigated. According to the theory of binary systems, a mathematical model of admixture diffusion in a multiphase body with spherical randomly disposed inclusions of different radii is constructed. The dense packing of spheres with different radii is used to modeling the skeleton of the body. The contact initial-boundary value problem is reduced to the mass transfer equation for the whole body. Its solution is constructed in the form of Neumann series. On the basis of the obtained calculation formula, a quantitative analysis of the mass transfer of admixture in the body with spherical inclusions, which are filled with materials of fundamentally different physical nature, but commensurable volume fractions, is carried out. It is shown that in modeling skeleton by spheres of one characteristic radius averaged concentration values coincide for different cases of radius, such as when characteristic radius equals to the average value of the radii of inclusions; or to the radius corresponding the smallest spherical inclusion; or to the radius of an order of magnitude smaller than this value.