MATHEMATICAL MODELLING OF ADMIXTURE CONCENTRATION DISTRIBUTION IN STOCHASTIC STRATIFIED BODIES UNDER NONIDEAL CONTACT CONDITIONS ON INTERPHASES MEASURES
Journal Title: Вісник Кременчуцького національного університету імені Михайла Остроградського - Year 2017, Vol 1, Issue 104
Abstract
Purpose. To investigate the processes of admixture diffusion in a two-phase stratified strip of randomly nonhomogeneous structure taking into account jump discontinuities of the function of concentration as well as its derivative on interphases. Methodology. We have applied mathematical modelling the diffusion processes in stochastic stratified bodies basing on the approach to mathematical description of mass transfer in randomly nonhomogeneous structures, in which sizes of nonhomogeneities can be commensurable with the body sizes. By this approach the random contact initial- boundary value problem of admixture mass transfer is reduced to the equivalent integro-differentual equation, and its solution is obtained in the form of Neumann series that gives the opportunity to perform of procedure of averaging the concentration field over the ensemble of phase configurations. Results. We have obtained the calculating formula for the admixture concentration field averaged over the ensemble of phase configurations with the function of uniform distribution. We have shown that the calculating formula for concentration at explicit accounting its jump discontinui-ties on interphases involves an additional term that does not change the behaviour of the concentration field under small volume fraction of inclusions but can greatly impact on its values. Numerical experiments have been carried out and we have established influence of the medium parameters on distributions of the concentration of admixture particles in sto-chastic stratified bodies under non-ideal contact conditions. We have brought to light three extremes in behavior of new summand of the averaged concentration. We have established the domain of the problem parameters in which this term is negligibly small. Originality. For the first time, in modelling and simulation of physical processes in multiphase ran-domly nonhomogeneous bodies we have proposed new presentation of the operator of diffusion equation for the whole body, which takes explicitly account of jump discontinuities of the sought function and its derivatives on random interfases. We have constructed new integro-differentual equation with random kernel that is equivalent to the original problem. This equation has been solved by the method of iterations and ts solution has been found in the form of Neu-mann series. We have formulated and proved the theorems of both convergence of the obtained series and existence of the solution of the integro-differentual equation. The way to prove the first theorem is also new and original, basing on determining the regularities for the coefficients of majorizations provided recursively. Practical value. The results ob-tained in the study give the opportunity to investigate quantitatively concentration of admixture migrationg in the two-phase body with randomly disposed layered inclusions under uniform distribution of the phases with explicit accounting jumps of the concentration and its derivative on interphases that is observed in experiments. On the basis on the ob-tained calculating formulae, the software has been created. The determined regularities can be used for creation of mate-rials with desired properties, at many technological operations, including alloy homogenization, metallization and weld of materials, especially for optimization of work of composite materials in the conditions of aggressive external medi-um, large external loads and temperatures that leads to changes in structure of multiphase materials, formation of seg-ments of the additional immiscibility of interphase damage and degradation of the functional properties of the material.
Authors and Affiliations
О. Chernukha, Y. Bilushchak, V. Goncharuk
Keywords
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