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Homogenization results for a class of parabolic equations with a non-local interface condition via time-periodic unfolding

  • M. AmarEmail author
  • D. Andreucci
  • R. Gianni
  • C. Timofte
Article
  • 33 Downloads

Abstract

We study the thermal properties of a composite material in which a periodic array of finely mixed perfect thermal conductors is inserted. The suitable model describing the behaviour of such physical materials leads to the so-called equivalued surface boundary value problem. To analyze the overall conductivity of the composite medium (when the size of the inclusions tends to zero), we make use of the homogenization theory, employing the unfolding technique. The peculiarity of the problem under investigation asks for a particular care in developing the unfolding procedure, giving rise to a non-standard two-scale problem.

Keywords

Homogenization Time-periodic unfolding Total flux boundary conditions Parabolic problems 

Mathematics Subject Classification

35B27 35Q79 35K20 

Notes

Acknowledgements

The first author is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The second author is member of the Gruppo Nazionale per la Fisica Matematica (GNFM) of the Istituto Nazionale di Alta Matematica (INdAM). The last author wishes to thank Dipartimento di Scienze di Base e Applicate per l’Ingegneria for the warm hospitality and Università “La Sapienza” of Rome for the financial support.

References

  1. 1.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23, 1482–1518 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Amar, M., Andreucci, D., Bellaveglia, D.: Homogenization of an alternating Robin–Neumann boundary condition via time-periodic unfolding. Nonlinear Anal. Theory Methods Appl. 153, 56–77 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Amar, M., Andreucci, D., Bellaveglia, D.: The time-periodic unfolding operator and applications to parabolic homogenization. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28, 663–700 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Amar, M., Andreucci, D., Gianni, R., Timofte, C.: Well-posedness of two pseudo-parabolic problems for electrical conduction in heterogenous media. Submitted (2019)Google Scholar
  5. 5.
    Amar, M., Andreucci, D., Gianni, R., Timofte, C.: Homogenization of a heat conduction problem with a total flux boundary condition. To appear in “Proceedings of XXIV AIMETA Conference 2019”, Lecture Notes in Mechanical Engineering, SpringerGoogle Scholar
  6. 6.
    Andreucci, D., Bellaveglia, D.: Permeability of interfaces with alternating pores in parabolic problems. Asymptot. Anal. 79, 189–227 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Andreucci, D., Bellaveglia, D., Cirillo, E.N.M.: A model for enhanced and selective transport through biological membranes with alternating pores. Math. Biosci. 257, 42–49 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Andreucci, D., Gianni, R.: Global existence and blow up in a problem with non local dynamical boundary conditions. Adv. Differ. Equ. 1, 729–752 (1996)zbMATHGoogle Scholar
  9. 9.
    Bellieud, M.: Homogenization of evolution problems for a composite medium with very small and heavy inclusions. ESAIM: COCV 11(2), 266–284 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bellieud, M.: Vibrations d’un composite élastique comportant des inclusions granulaires très lourdes : effets de mémoire. C. R. Math. 346(13), 807–812 (2008)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bellieud, M.: Torsion effects in elastic composites with high contrast. SIAM J. Math. Anal. 41(6):2514–2553 (2009/2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bellieud, M., Gruais, I.: Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non-local effects. Memory effects. J. Math. Pures Appl. 84(1), 55–96 (2005)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Briane, M.: Homogenization of the torsion problem and the Neumann problem in nonregular periodically perforated domains. Math. Models Methods Appl. Sci. (6) 7, 847–870 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Cioranescu, D., Damlamian, A., Donato, P., Griso, G., Zaki, R.: The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44(2), 718–760 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. 335(1), 99–104 (2002)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Cioranescu, D., Damlamian, A., Li, T.: Periodic homogenization for inner boundary conditions with equi-valued surfaces: the unfolding approach. Chin. Ann. Math. Ser. B 34B(2), 213–236 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cioranescu, D., Donato, P., Zaki, R.: Periodic unfolding and Robin problems in perforated domains. C. R. Math. 342(1), 469–474 (2006)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Cioranescu, D., Donato, P., Zaki, R.: The periodic unfolding method in perforated domains. Port. Math. 63(4), 467–496 (2006)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Cioranescu, D., Jean Paulin, J.Saint: Homogenization in open sets with holes. J. Math. Anal. Appl. (2) 71, 590–607 (1979)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ebadi-Dehaghani, H., Nazempour, M.: Thermal conductivity of nanoparticles filled polymers. Smart Nanopart. Technol. 23, 519–540 (2012)Google Scholar
  22. 22.
    Gorb, Y., Berlyand, L.: Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape. Q. J. Mech. Appl. Math. 1(58), 83–106 (2005)CrossRefGoogle Scholar
  23. 23.
    Kemaloglu, S., Ozkoc, G., Aytac, A.: Thermally conductive boron nitride/sebs/eva ternary composites:processing and characterisation. Polymer Composites (Published online on www.interscience. wiley.com, 2009, Society of Plastic Engineers), pp. 1398–1408 (2010)Google Scholar
  24. 24.
    Li, F.: Existence and uniqueness of bounded weak solution for non-linear parabolic boundary value problem with equivalued surface. Math. Methods Appl. Sci. 27, 1115–1124 (2004)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Li, T.: A class of non-local boundary value problems for partial differential equations and its applications in numerical analysis. J. Comput. Appl. Math. 28, 49–62 (1989)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, T., Zheng, S., Tan, Y., Shen, W.: Boundary Value Problems with Equivalued Surfaces and Resistivity Well-Logging. Pitman Research Notes in Mathematics Series, vol. 382. Longman, Harlow (1998)zbMATHGoogle Scholar
  27. 27.
    Phromma, W., Pongpilaipruet, A., Macaraphan, R.: Preparation and thermal properties of PLA filled with natural rubber-PMA core-shell/magnetite nanoparticles. In: European Conference; 3rd, Chemical Engineering. Recent Advances in Engineering, Paris (2012)Google Scholar
  28. 28.
    Shahil, K., Balandin, A.: Graphene-based nanocomposites as highly efficient thermal interface materials. In: Graphene Based Thermal Interface Materials, pp. 1–18 (2011)Google Scholar
  29. 29.
    Yang, X., Liang, C., Ma, T., Guo, Y., Kong, J., Gu, J., Chen, M., Zhu, J.: A review on thermally conductive polymeric composites: classification, measurement, model and equations, mechanism and fabrication methods. Adv. Compos. Hybrid Mater. 1, 207–230 (2018)CrossRefGoogle Scholar
  30. 30.
    Zhang, L., Deng, H., Fu, Q.: Recent progress on thermal conductive and electrical insulating polymer composites. Compos. Commun. 8, 74–82 (2018)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze di Base e Applicate per l’IngegneriaSapienza - Università di RomaRomeItaly
  2. 2.Dipartimento di Matematica ed InformaticaUniversità di FirenzeFlorenceItaly
  3. 3.Faculty of PhysicsUniversity of BucharestBucharestRomania

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