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June  2018, 23(4): 1739-1756. doi: 10.3934/dcdsb.2018078

## Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices

 1 Dipartimento di Scienze di Base e Applicate per l'Ingegneria, Sapienza Università di Roma, Via A. Scarpa 16, 00161 Roma, Italy 2 Dipartimento di Matematica ed Informatica, Università di Firenze, Via Santa Marta 3, 50139 Firenze, Italy

* Corresponding author

Received  June 2016 Revised  July 2017 Published  March 2018

In this paper we study a model for the heat conduction in a composite having a microscopic structure arranged in a periodic array. We obtain the macroscopic behaviour of the material and specifically the overall conductivity via an homogenization procedure, providing the equation satisfied by the effective temperature.

Citation: Micol Amar, Roberto Gianni. Laplace-Beltrami operator for the heat conduction in polymer coating of electronic devices. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1739-1756. doi: 10.3934/dcdsb.2018078
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, (1995), 15-25. [3] G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA Journal of Applied Mathematics, 77 (2012), 788-815. doi: 10.1093/imamat/hxs049. [4] G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis, 7 (1993), 81-95. [5] M. Amar, D. Andreucci and D. Bellaveglia, Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding, Nonlinear Analysis: Theory, Methods and Applications, 153 (2017), 56-77. doi: 10.1016/j.na.2016.05.018. [6] M. Amar, D. Andreucci and D. Bellaveglia, The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 663-700. doi: 10.4171/RLM/781. [7] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1261-1295. doi: 10.1142/S0218202504003623. [8] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787. doi: 10.1002/mma.709. [9] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues, Euro. Jnl. of Applied Mathematics, 20 (2009), 431-459. doi: 10.1017/S0956792509990052. [10] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues, J. Mechanics of Material and Structures, 4 (2009), 211-223. doi: 10.2140/jomms.2009.4.211. [11] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range, Communications on Pure and Applied Analysis, 9 (2010), 1131-1160. doi: 10.3934/cpaa.2010.9.1131. [12] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differential and Integral Equations, 26 (2013), 885-912. [13] M. Amar and R. Gianni, Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator, (2018), submitted. [14] M. Amar and R. Gianni, Error estimate for a homogenization problem involving the Laplace-Beltrami operator, Mathematics and Mechanics of Complex Systems (2018), to appear, arXiv: 1705.04345v2. [15] M. Amar and R. Gianni, Existence and uniqueness for a two-scale system involving tangential operators, (2018), work in progress. [16] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM Journal on Mathematical Analysis, 44 (2012), 718-760. doi: 10.1137/100817942. [17] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, Comptes Rendus Mathematique, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9. [18] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM Journal on Mathematical Analysis, 40 (2008), 1585-1620. doi: 10.1137/080713148. [19] D. Cioranescu, P. Donato and R. Zaki, Periodic unfolding and robin problems in perforated domains, Comptes Rendus Mathematique, 342 (2006), 469-474. doi: 10.1016/j.crma.2006.01.028. [20] D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496. [21] P. Donato and Z. Yang, The periodic unfolding method for the wave equation in domains with holes, Adv. Math.Sci. Appl., 22 (2012), 521-551. [22] P. Donato and Z. Yang, The periodic unfolding method for the heat equation in perforated domains, Science China Mathematics, 59 (2016), 891-906. doi: 10.1007/s11425-015-5103-4. [23] H. Ebadi-Dehaghani and M. Nazempour, Thermal conductivity of nanoparticles filled polymers, Smart Nanoparticles Technology, 23 (2012), 519-540. [24] L. Flodén, A. Holmbom, M. Olsson and J. Persson, Very weak multiscale convergence, Applied Mathematics Letters, 23 (2010), 1170-1173. doi: 10.1016/j.aml.2010.05.005. [25] L. Flodén, A. Holmbom and M. Olsson Lindberg, A strange term in the homogenization of parabolic equations with two spatial and two temporal scales, Journal of Function Spaces and Applications (2012), Art. ID 643458, 9 pp. [26] A. Holmbom, Homogenization of parabolic equations an alternative approach and some corrector-type results, Applications of Mathematics, 42 (1997), 321-343. doi: 10.1023/A:1023049608047. [27] S. Kemaloglu, G. Ozkoc and A. Aytac, Thermally conductive boron nitride/sebs/eva ternary composites: processing and characterization, Polymer Composites (Published online on http://onlinelibrary.wiley.com/doi/10.1002/pc.20925/full, 2009, Society of Plastic Engineers), (2010), 1398–1408. [28] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [29] G. Nguetseng and J. Woukeng, Σ-convergence of nonlinear parabolic operators, Nonlinear Analysis, 66 (2007), 968-1004. doi: 10.1016/j.na.2005.12.035. [30] W. Phromma, A. Pongpilaipruet and R. Macaraphan, Preparation and thermal properties of PLA filled with natural rubber-PMA core-shell/magnetite nanoparticles, European Conference; 3rd, Chemical Engineering, Recent Advances in Engineering. Paris, (2012). [31] K. M. Shahil and A. A. Balandin, Graphene-based nanocomposites as highly efficient thermal interface materials, Graphene Based Thermal Interface Materials, (2011), 1-18. [32] V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sbornik: Mathematics, 191 (2000), 973-1014.

show all references

##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] G. Allaire, A. Damlamian and U. Hornung, Two-scale convergence on periodic surfaces and applications, Proceedings of the International Conference on Mathematical Modelling of Flow through Porous Media, (1995), 15-25. [3] G. Allaire and H. Hutridurga, Homogenization of reactive flows in porous media and competition between bulk and surface diffusion, IMA Journal of Applied Mathematics, 77 (2012), 788-815. doi: 10.1093/imamat/hxs049. [4] G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis, 7 (1993), 81-95. [5] M. Amar, D. Andreucci and D. Bellaveglia, Homogenization of an alternating Robin-Neumann boundary condition via time-periodic unfolding, Nonlinear Analysis: Theory, Methods and Applications, 153 (2017), 56-77. doi: 10.1016/j.na.2016.05.018. [6] M. Amar, D. Andreucci and D. Bellaveglia, The time-periodic unfolding operator and applications to parabolic homogenization, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 28 (2017), 663-700. doi: 10.4171/RLM/781. [7] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Mathematical Models and Methods in Applied Sciences, 14 (2004), 1261-1295. doi: 10.1142/S0218202504003623. [8] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, On a hierarchy of models for electrical conduction in biological tissues, Mathematical Methods in the Applied Sciences, 29 (2006), 767-787. doi: 10.1002/mma.709. [9] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Exponential asymptotic stability for an elliptic equation with memory arising in electrical conduction in biological tissues, Euro. Jnl. of Applied Mathematics, 20 (2009), 431-459. doi: 10.1017/S0956792509990052. [10] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Stability and memory effects in a homogenized model governing the electrical conduction in biological tissues, J. Mechanics of Material and Structures, 4 (2009), 211-223. doi: 10.2140/jomms.2009.4.211. [11] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, Homogenization limit and asymptotic decay for electrical conduction in biological tissues in the high radiofrequency range, Communications on Pure and Applied Analysis, 9 (2010), 1131-1160. doi: 10.3934/cpaa.2010.9.1131. [12] M. Amar, D. Andreucci, P. Bisegna and R. Gianni, A hierarchy of models for the electrical conduction in biological tissues via two-scale convergence: The nonlinear case, Differential and Integral Equations, 26 (2013), 885-912. [13] M. Amar and R. Gianni, Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator, (2018), submitted. [14] M. Amar and R. Gianni, Error estimate for a homogenization problem involving the Laplace-Beltrami operator, Mathematics and Mechanics of Complex Systems (2018), to appear, arXiv: 1705.04345v2. [15] M. Amar and R. Gianni, Existence and uniqueness for a two-scale system involving tangential operators, (2018), work in progress. [16] D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM Journal on Mathematical Analysis, 44 (2012), 718-760. doi: 10.1137/100817942. [17] D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization, Comptes Rendus Mathematique, 335 (2002), 99-104. doi: 10.1016/S1631-073X(02)02429-9. [18] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM Journal on Mathematical Analysis, 40 (2008), 1585-1620. doi: 10.1137/080713148. [19] D. Cioranescu, P. Donato and R. Zaki, Periodic unfolding and robin problems in perforated domains, Comptes Rendus Mathematique, 342 (2006), 469-474. doi: 10.1016/j.crma.2006.01.028. [20] D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496. [21] P. Donato and Z. Yang, The periodic unfolding method for the wave equation in domains with holes, Adv. Math.Sci. Appl., 22 (2012), 521-551. [22] P. Donato and Z. Yang, The periodic unfolding method for the heat equation in perforated domains, Science China Mathematics, 59 (2016), 891-906. doi: 10.1007/s11425-015-5103-4. [23] H. Ebadi-Dehaghani and M. Nazempour, Thermal conductivity of nanoparticles filled polymers, Smart Nanoparticles Technology, 23 (2012), 519-540. [24] L. Flodén, A. Holmbom, M. Olsson and J. Persson, Very weak multiscale convergence, Applied Mathematics Letters, 23 (2010), 1170-1173. doi: 10.1016/j.aml.2010.05.005. [25] L. Flodén, A. Holmbom and M. Olsson Lindberg, A strange term in the homogenization of parabolic equations with two spatial and two temporal scales, Journal of Function Spaces and Applications (2012), Art. ID 643458, 9 pp. [26] A. Holmbom, Homogenization of parabolic equations an alternative approach and some corrector-type results, Applications of Mathematics, 42 (1997), 321-343. doi: 10.1023/A:1023049608047. [27] S. Kemaloglu, G. Ozkoc and A. Aytac, Thermally conductive boron nitride/sebs/eva ternary composites: processing and characterization, Polymer Composites (Published online on http://onlinelibrary.wiley.com/doi/10.1002/pc.20925/full, 2009, Society of Plastic Engineers), (2010), 1398–1408. [28] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [29] G. Nguetseng and J. Woukeng, Σ-convergence of nonlinear parabolic operators, Nonlinear Analysis, 66 (2007), 968-1004. doi: 10.1016/j.na.2005.12.035. [30] W. Phromma, A. Pongpilaipruet and R. Macaraphan, Preparation and thermal properties of PLA filled with natural rubber-PMA core-shell/magnetite nanoparticles, European Conference; 3rd, Chemical Engineering, Recent Advances in Engineering. Paris, (2012). [31] K. M. Shahil and A. A. Balandin, Graphene-based nanocomposites as highly efficient thermal interface materials, Graphene Based Thermal Interface Materials, (2011), 1-18. [32] V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sbornik: Mathematics, 191 (2000), 973-1014.
Left: the periodic cell $Y$. ${E_{{\rm{int}}}}$ is the shaded region and ${E_{{\rm{out}}}}$ is the white region. Right: the region
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