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Mathematical modeling and simulation of diffusion processes in stochastically nonhomogeneous stratified structures

Work number - M 33

Presented by Centre of Mathematical Modelling of Ya.S.Pidstryhach Institute of Applied Problems of Mechanics and Mathematics of NAS of Ukraine

Authors: Bilushchak Yurii Igorovych, Chuchvara Anastasiia Yevhenivna

The goal of the work is creation new approach to mathematical description of the mass transfer processes in randomly nonhomogeneous stratified bodies with arbitrary sizes of inclusions at any probable distribution; within the approach to develop methodologies of study of random fields of concentration under non-ideal contact conditions, stochastical diffusion fluxes in stratified structures as well as dispersion and two-point function of correlation (autocorrelation) of the field of concentration of migrating substance.

A new approach to description of diffusion processes in two-phase randomly nonhomo­geneous bodies is proposed. It is based on use of the generalized functions, integral equa­tions, probability theory and the method of Green functions. The approach empowers to de­termine averaged over the ensemble of phase configurations fields of concentration and flows of migrating substance taking into account essentially different diffusive properties of the phases, arbitrary sizes of inclusions and arbitrary probabilistic distribution of layers. A new equation of mass transfer is obtained for a two-phase body that takes explicitly into account jump discontinuity of the sought function and equality of the flows on the phase contact boundaries. The methodology is proposed and justified for mathematical description of field dispersion and correlation function of the field of concentration of substance diffu­sing in two-phase randomly nonhomogeneous stratified bodies, which uses presentation of the concentration field in terms of convergent integral Neumann series and considers ave­raging over the ensemble of phase configurations. On the basis of the relationship of mass balance a new differential equation is obtained for the function of diffusion flux of admixture particles, where nonhomogeneity of material structure is considered into the equation coefficients; initial and boundary conditions for the flow are justified. It is constructed the integro-differential equation for the function of mass flow, which is equivalent the original initial-boundary value problem. Its solution is obtained in terms of Neumann series. The obtained calculating formulae are realized as package software «Ro-conc» and«FlowRan».

The results were used for calculation of distributions of carbon and hydrogen in composite material steel 38HNZMFA-Ni; for studying loss of functional properties of building structures, in particular, welds, panels and blocks as a result of oxygen diffusion from atmosphere and further oxidation of structural metal during designing clad reinforced concrete structures; for estimation of both time and efficiency of operation of water filters.

The results of researchwork contributes into solving the problem of mathematical modeling non-equilibrium processes in media of complex inner structure. They can be applied into geophysics, ecology, building branch, microelectronics, mechanical engineering, space and aircraft industry.

The new developed approaches to description of varied aspects of diffusion processes are new. And the methodology for determining the second moments of a random field as well as the obtained equation of diffusion for mass fluxes have neither domestic nor foreign analogs.

Number of publicationsis34, including 1 chapter of monograph, 27 papers(3 in fo­reignjournals). According to the Google Scholar the total number of referencesis 46, h-index (for the work) = 5.Novelty and competitiveness oftechnical decisions areprotected by two certificates of registration of copyrightsfor packageof computer programs.