Mathematical Modelling of Diffusion Flows in Two-Phase Stratified Bodies with Randomly Disposed Layers of Stochastically Set Thickness

Abstract

The work is dedicated to mathematical modelling of random diffusion flows of admixture particles in a two-phase stratified strip with stochastic disposition of phases and random thickness of inclusion-layers. The study of such models are especially important during the creation of composite layered materials, in the research of the transmission properties of filters, and in the prediction of the spread of pollutants in the environment. Within the model we consider one case of uniform distribution of coordinates of upper boundaries of the layers of which the body is made up and two more cases, i.e., of uniform and triangular distributions of the inclusion thickness. The initial-boundary value problems of diffusion are formulated for flux functions; the boundary conditions at one of the body’s surfaces are set for flux and, at the other boundary, the conditions are given for admixture concentration; the initial condition being concerned with zero and non-zero constant initial concentrations. An equivalent integro-differential equation is constructed. Its solution is found in terms of Neumann series. For the first time it was obtained calculation formulae for diffusion flux averaged over the ensemble of phase configurations and over the inclusion thickness. It allowed to investigate the dependence of averaged diffusion fluxes on the medium’s characteristics on the basis of the developed software. The simulation of averaged fluxes of admixture in multilayered Fe−Cu and αFe−Ni materials is made. Comparative analysis of solutions, depending on the stage of averaging procedure over thickness, is carried out. It is shown that for some values of parameters the stage of averaging procedure over thickness has almost no effect on the diffusion flow value.

Keywords

• mathematical modeling
• diffusion flow
• stochastic stratified structure
• integro-differential equation
• Neumann series
• random inclusion thickness