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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

Micol Amar 1,, and  Roberto Gianni 2,

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.

Keywords: Homogenization, unfolding periodic method, Laplace-Beltrami operator, tangential derivatives, partial differential equations, polymer encapsulation.
Mathematics Subject Classification: Primary: 35B27; Secondary: 35Q79, 35K20.
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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis, 7 (1993), 81-95.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

M. Amar and R. Gianni, Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator, (2018), submitted. Google Scholar

[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. Google Scholar

[15]

M. Amar and R. Gianni, Existence and uniqueness for a two-scale system involving tangential operators, (2018), work in progress. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

H. Ebadi-Dehaghani and M. Nazempour, Thermal conductivity of nanoparticles filled polymers, Smart Nanoparticles Technology, 23 (2012), 519-540.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.   Google Scholar

[32]

V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sbornik: Mathematics, 191 (2000), 973-1014.   Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated holes, Asymptotic Analysis, 7 (1993), 81-95.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[13]

M. Amar and R. Gianni, Existence, uniqueness and concentration for a system of PDEs involving the Laplace-Beltrami operator, (2018), submitted. Google Scholar

[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. Google Scholar

[15]

M. Amar and R. Gianni, Existence and uniqueness for a two-scale system involving tangential operators, (2018), work in progress. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

D. Cioranescu, P. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[23]

H. Ebadi-Dehaghani and M. Nazempour, Thermal conductivity of nanoparticles filled polymers, Smart Nanoparticles Technology, 23 (2012), 519-540.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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). Google Scholar

[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.   Google Scholar

[32]

V. Zhikov, On an extension of the method of two-scale convergence and its applications, Sbornik: Mathematics, 191 (2000), 973-1014.   Google Scholar

Figure 1.  Left: the periodic cell $Y$. ${E_{{\rm{int}}}}$ is the shaded region and ${E_{{\rm{out}}}}$ is the white region. Right: the region Ω
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