Journal of Mathematical Sciences

, Volume 240, Issue 1, pp 70–85 | Cite as

Mathematical Modeling of the Mean Concentration Field in Random Stratified Structures with Regard for the Jumps of Sought Function on the Interfaces

  • О. Yu. Chernukha
  • Yu. I. Bilushchak

We study the diffusion processes of an admixture in a two-phase stratified strip of randomly inhomogeneous structure with regard for the jumps of the concentration function and its derivative on the contact boundaries of the phases. A new representation of the operator of equation of mass transfer for the entire body is proposed. We formulate an equivalent integrodifferential equation whose solution is constructed in the form of a Neumann integral series. The obtained solution is averaged over the ensemble of phase configurations with uniform distribution function. It is shown that the computational formula for the mean concentration with explicit account of its jumps on the interfaces contains an additional term. It is demonstrated that the ratios of the diffusion coefficients, the concentration dependences of the chemical potentials in different phases, and their relationships affect the sign of this term. We find the ranges of parameters of the problem for which this term is negligibly small.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ya. Yo. Burak, Ye. Ya. Chaplya, and O. Yu. Chernukha, Continual Thermodynamic Models of Mechanics of Solid Solutions [in Ukrainian], Naukova Dumka, Kyiv (2006).Google Scholar
  2. 2.
    V. S. Korolyuk, N. I. Portenko, A. V. Skorokhod, and A. F. Turbin, A Handbook of the Theory of Probabilities and Mathematical Statistics [in Russian], Nauka, Moscow (1985).Google Scholar
  3. 3.
    M. L. Krasnov, Integral Equations. Introduction to the Theory [in Russian], Nauka, Moscow (1975).Google Scholar
  4. 4.
    V. V. Lytvyn, D. I. Uhryn, and A. M. Fit’o, “Modeling of the process of formation of territorial communities as problems of partition of graphs,” Skhid.-Evrop. Zh. Pered. Tekhnol., 1, No. 4 (79), 47–52 (2016).Google Scholar
  5. 5.
    A. V. Luikov, Analytical Heat Diffusion Theory, Vysshaya Shkola, Moscow (1967); English translation: Academic Press, New York (2012).Google Scholar
  6. 6.
    A. Münster, Chemische Thermodynamik, Verlag Chemie, Berlin (1969).Google Scholar
  7. 7.
    S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Introduction to Statistical Radiophysics, Vol. 2: Random Fields [in Russian], Nauka, Moscow (1978).Google Scholar
  8. 8.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1972).Google Scholar
  9. 9.
    L. P. Khoroshun, “Mathematical models and methods of the mechanics of stochastic composites,” Prikl. Mekh., 36, No. 10, 30–62 (2000); English translation: Int. Appl. Mech., 36, No. 10, 1284–1316 (2000).Google Scholar
  10. 10.
    Ye. Ya. Chaplya and O. Yu. Chernukha, Mathematical Modeling of Diffusion Processes in Random and Regular Structures [in Ukrainian], Naukova Dumka, Kyiv (2009).Google Scholar
  11. 11.
    Ye. Ya. Chaplya, O. Yu. Chernukha, and Yu. I. Bilushchak, “Contact initial boundary-value problem of the diffusion of admixture particles in a two-phase stochastically inhomogeneous stratified strip,” Mat. Met. Fiz.-Mekh. Polya, 54, No. 1, 79–90 (2011); English translation: J. Math. Sci., 183, No. 1, 83–99 (2012).Google Scholar
  12. 12.
    O. Yu. Chernukha, Yu. I. Bilushchak, and A. E. Chuchvara, Modeling of Diffusion Processes in Stochastically Inhomogeneous Layered Structures [in Ukrainian], Rastr-7, Lviv (2016).Google Scholar
  13. 13.
    G. P. Chaudhari and V. Acoff, “Cold roll bonding of multi-layered bi-metal laminate composites,” Compos. Sci. Technol., 69, No. 10, 1667–1675 (2009).CrossRefGoogle Scholar
  14. 14.
    M. V. Davydov, “A probabilistic search algorithm for finding suboptimal branchings in mutually exclusive hypothesis graph,” Int. J. Knowl.-Based Intell. Eng. Systems., 18, No. 4, 247–253 (2014).CrossRefGoogle Scholar
  15. 15.
    J. B. Keller, “Flow in random porous media,” Transp. Porous Media, 43, No. 3, 395–406 (2001).MathSciNetCrossRefGoogle Scholar
  16. 16.
    T. Loimer and P. Uchytil, “Influence of the flow direction on the mass transport of vapors through membranes consisting of several layers,” Exp. Therm. Fluid Sci., 67, 2–5 (2015).CrossRefGoogle Scholar
  17. 17.
    A. Mikdam, A. Makradi, S. Ahzi, H. Garmestani, D. S. Li, and Y. Remond, “Effective conductivity in isotropic heterogeneous media using a strong-contrast statistical continuum theory,” J. Mech. Phys. Solids, 57, No. 1, 76–86 (2009).CrossRefzbMATHGoogle Scholar
  18. 18.
    A. H. W. Ngan, “Canonical ensemble for static elastic structures with random microstructures,” J. Mech. Phys. Solids, 57, No. 5, 803–811 (2009).CrossRefGoogle Scholar
  19. 19.
    Y. Yang, D. Wang, J. Lin, D. F. Khan, G. Lin, and J. Ma, “Evolution of structure and fabrication of Cu/Fe multilayered composites by a repeated diffusion-rolling procedure,” Mater. Design, 85, 635–639 (2015).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • О. Yu. Chernukha
    • 1
  • Yu. I. Bilushchak
    • 1
  1. 1.Center of Mathematical Modeling, Pidstryhach Institute for Applied Problems in Mechanics and MathematicsUkrainian National Academy of SciencesLvivUkraine

Personalised recommendations