517.958:532.72

mathematical modeling, multiphase stochastic structure, spherical inclusion, admixture concentration, dense packing of spheres, averaging over the ensemble of phase configurations, computing.

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.

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1. Coutelieris A. F., Delgado J. M. P. Q. Transport Processes in Porous Media, Berlin, Springer, 2012, 235 p.
2. Kovbashyn V., Bochar I. The study of technologies to improve physical-mechanical and chemical properties of reaction sintered ceramic materials on the basis of silicon carbide. Scientific Journal of TNTU. Vol. 86. No. 2. 2017. P. 14–20.
3. Van Kampen N. G. Stochastic Processes in Chemistry and Physics, Norwell, Elsevier, 1992, 480 p.
4. Vamoş C., Suciua N., Vereecken H. Generalized random walk algorithm for the numerical modeling of complex diffusion processes, Journal of Computational Physics, Vol. 186 (2). 2003. Р. 527–544.
5. LaBolle E. M., Quastel J., Fogg E. G., Gravner J. Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients. Water Resources Research. Vol. 36 (3). 2000. Р. 651–662.
6. Yong Y., Lou X., Li S., Yang C., Yin X. Direct simulation of the influence of the pore structure on the diffusion process in porous media, Computers & Mathematics with Applications. Vol. 67 (2). 2014.
   Р. 412–423.
7. Hlushkou V., Khirevich S., Apanasovich V.V., Tallarek U. Pore-scale dispersion in electrokinetic flow through a random sphere packing, Analytical Chemistry. 79. 2007. Р. 113-121.
8. Chaplia Y., Chernukha O. Three-dimensional diffusion in a multiphase body with randomly disposed inclusions of a spherical form, International Journal of Heat and Mass Transfer, Vol. 46. 2003. Р. 3323–3328.
9. Chaplia Ye., Chernukha O. Fizyko-matematychne modeliuvannia dyfuziinykh protsesiv u vypadkovykh i rehuliarnykh strukturakh. Kyiv: Naukova dumka, 2009. 302 p. [In Ukrainian].
10. Chernukha O., Bilushchak Yu., Chuchvara A. Modeliuvannia dyfuziinykh protsesiv u stokhastychno neodnoridnykh strukturakh. Lviv: Rastr-7, 2016. 262 p. [In Ukrainian].
11. Gyarmati J. Non-equilibrium Thermodynamics. Field Theory and Variational Principles. Berlin, Springer, 1970, 184 p.
12. Münster A. Classical Thermodynamics. Wiley Interscience, 1970, 387 p.
13. Chernukha O., Chuchvara A. Modeliuvannia dyfuzii domishkovoi rechovyny u porystomu tili z vypadkovymy sferychnymy poramy pry sumirnykh obiemnykh chastkakh faz. Mat. metody ta fiz.-mekh. polia. 62. № 1. 2019. Р. 150–161. [In Ukrainian].
14. Rytov S. M., Kravtsov Yu. A., Tatarskyi V. Y. Vvedenie v statisticheskuiu radiofiziku II. Sluchainye polia. Moskva: Nauka, 1978. 464 p. [Іn Russian].
15. Spravochnik po spetsyalnym funktsyiam / pod red. M. Abramovitsa i Y. Styhan. Moskva: Mir, 1979, 830 p. [In Russian].
16. Lykov A. V. Teoriia teploprovodnosti. Moskva: Vysshaia shkola, 1978. 480 p. [In Russian].