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.

Some Remarks on Smooth Mappings of Hilbert and Banach Spaces and Their Local Convexity Property

“…In particular, in [7][8][9], the convexity property was effectively applied toward solving existence problems for both functional-difference inclusions and kinetic equations of statistical physics. It proved to be especially fruitful for the theory of nonlinear differential-operator equations [10][11][12], control theory, and optimization theory [13,14]. Some interesting and important local convexity properties, relevant to mappings of Hilbert spaces, were initially discussed in [3], and later generalized and studied in [15][16][17], devoted to the closedness of quadratic mappings on a separable Hilbert space.…”

The Modeling of Self-Consistent Electron–Deformation–Diffusion Effects in Thin Films with Lattice Parameter Mismatch

“…However, this model does not take into account the effects caused by the non-local interaction between defects and matrix atoms. The authors of [13] developed the mathematical model of diffusion in layered materials, which can be used for thin films. However, this model does not take into account the influence of deformation caused by the non-zero volume of defects on their diffusion.…”