American Institute of Mathematical Sciences

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December  2016, 11(4): 545-562. doi: 10.3934/nhm.2016009

Homogenization of a thermal problem with flux jump

Renata Bunoiu 1, and  Claudia Timofte 2,

1. 

Institut Élie Cartan de Lorraine, CNRS, UMR 7502, Université de Lorraine, Metz, 57045, France
2. 

University of Bucharest, Faculty of Physics, Bucharest-Magurele, P.O. Box MG-11, Romania

Received  December 2015 Published  October 2016

The goal of this paper is to analyze, through homogenization techniques, the effective thermal transfer in a periodic composite material formed by two constituents, separated by an imperfect interface where both the temperature and the flux exhibit jumps. Following the hypotheses on the flux jump, two different homogenized problems are obtained. These problems capture in various ways the influence of the jumps: in the homogenized coefficients, in the right-hand side of the homogenized problem, and in the correctors.
Keywords: the periodic unfolding method., imperfect interface, Homogenization.
Mathematics Subject Classification: Primary: 35B27, 80M35; Secondary: 80M4.
Citation: Renata Bunoiu, Claudia Timofte. Homogenization of a thermal problem with flux jump. Networks & Heterogeneous Media, 2016, 11 (4) : 545-562. doi: 10.3934/nhm.2016009
References:
[1]

M. Amar, D. Andreucci and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Math. Model. Methods Appl. Sci., 14 (2004), 1261-1295. doi: 10.1142/S0218202504003623.  Google Scholar

[2]

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, Differ. Integral Equations, 26 (2013), 885-912.  Google Scholar

[3]

J. L. Auriault, C. Boutin and C. Geindreau, Homogenization of Coupled Phenomena in Heterogenous Media, Wiley, 2010. doi: 10.1002/9780470612033.  Google Scholar

[4]

J. L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Int. J. of Heat and Mass Transfer, 37 (1994), 2885-2892. doi: 10.1016/0017-9310(94)90342-5.  Google Scholar

[5]

A. G. Belyaev, A. L. Pyatnitskiĭ and G. A. Chechkin, Averaging in a perforated domain with an oscillating third boundary condition, Sbornik: Mathematics, 192 (2001), 933-949. doi: 10.1070/SM2001v192n07ABEH000576.  Google Scholar

[6]

Y. Benveniste and T. Miloh, Imperfect soft and stiff interfaces in two-dimensional elasticity, Mech. Mater., 33 (2001), 309-323. doi: 10.1016/S0167-6636(01)00055-2.  Google Scholar

[7]

D. Brinkman, K. Fellner, P. Markowich and M. T. Wolfram, A drift-diffusion-reaction model for excitonic photovoltaic bilayers: Asymptotic analysis and a 2-D HDG finite element scheme, Math. Models Methods Appl. Sci., 23 (2013), 839-872. doi: 10.1142/S0218202512500625.  Google Scholar

[8]

R. Bunoiu and C. Timofte, On the homogenization of a two-permeability problem with flux jump,, work in progress., ().   Google Scholar

[9]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760. doi: 10.1137/100817942.  Google Scholar

[10]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. doi: 10.1137/080713148.  Google Scholar

[11]

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

[12]

D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal., 53 (2007), 209-235.  Google Scholar

[13]

I. Chourabi and P. Donato, Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptot. Anal., 92 (2015), 1-43.  Google Scholar

[14]

P. Donato, L. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: a memory effect, J. Math. Pures Appl., 87 (2007), 119-143. doi: 10.1016/j.matpur.2006.11.004.  Google Scholar

[15]

P. Donato and K. H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, Nonlinear Differ. Equ. Appl., 22 (2015), 1345-1380. doi: 10.1007/s00030-015-0325-2.  Google Scholar

[16]

P. Donato, K. H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J. Math. Sci. (N. Y.), 176 (2011), 891-927. doi: 10.1007/s10958-011-0443-2.  Google Scholar

[17]

P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance, Analysis and Applications, 2 (2004), 247-273. doi: 10.1142/S0219530504000345.  Google Scholar

[18]

P. Donato and I. Ţenţea, Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method, Boundary Value Problems, 2013 (2013), 14pp.  Google Scholar

[19]

H. I. Ene and D. Poliševski, Model of diffusion in partially fissured media, Z. Angew. Math. Phys., 53 (2002), 1052-1059. doi: 10.1007/PL00013849.  Google Scholar

[20]

H. I. Ene and C. Timofte, Microstructure models for composites with imperfect interface via the periodic unfolding method, Asymptot. Anal., 89 (2014), 111-122.  Google Scholar

[21]

H. I. Ene, C. Timofte and I. Ţenţea, Homogenization of a thermoelasticity model for a composite with imperfect interface, Bull. Math. Soc. Sci. Math. Roumanie, 58 (2015), 147-160.  Google Scholar

[22]

H. I. Ene and C. Timofte, Homogenization results for a dynamic coupled thermoelasticity problem, Romanian Reports in Physics, 68 (2016), 979-989. Google Scholar

[23]

K. Fellner and V. Kovtunenko, A discontinuous Poisson-Boltzmann equation with interfacial transfer: Homogenisation and residual error estimate, Applicable Analysis, (2015), 1-22. doi: 10.1080/00036811.2015.1105962.  Google Scholar

[24]

M. Gahn, P. Knabner and M. Neuss-Radu, Homogenization of reaction-diffusion processes in a two-component porous medium with a nonlinear flux condition at the interface, and application to metabolic processes in cells, SIAM J. Appl. Math., 76 (2016), 1819-1843. doi: 10.1137/15M1018484.  Google Scholar

[25]

Z. Hashin, Thin interphase-imperfect interface in elasticity with application to coated fiber composites, Journal of the Mechanics and Physics of Solids, 50 (2002), 2509-2537. doi: 10.1016/S0022-5096(02)00050-9.  Google Scholar

[26]

H. K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances, Appl. Anal., 75 (2000), 403-424. doi: 10.1080/00036810008840857.  Google Scholar

[27]

E. R. Ijioma, A. Muntean and T. Ogawa, Pattern formation in reverse smouldering combustion: A homogenization approach, Combustion Theory and Modelling, 17 (2013), 185-223. doi: 10.1080/13647830.2012.734860.  Google Scholar

[28]

E. C. Jose, Homogenization of a parabolic problem with an imperfect interface, Rev. Roum. Math. Pures Appl., 54 (2009), 189-222.  Google Scholar

[29]

A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000. Google Scholar

[30]

K. H. Le Nguyen, Homogenization of heat transfer process in composite materials, JEPE, 1 (2015), 175-188.  Google Scholar

[31]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.  Google Scholar

[32]

D. Polisevski, R. Schiltz-Bunoiu and A. Stanescu, Homogenization cases of heat transfer in structures with interfacial barriers, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 58 (2015), 463-473.  Google Scholar

[33]

D. Polisevski and R. Schiltz-Bunoiu, Heat conduction through a first-order jump interface, New Trends in Continuum Mechanics (M. Mihailescu-Suliciu ed.), Theta Series in Advanced Mathematics, 3 (2005), 225-230.  Google Scholar

[34]

D. Polisevski and R. Schiltz-Bunoiu, Diffusion in an intermediate model of fractured porous media, Bulletin Scientifique, Mathématiques et Informatique, 10 (2004), 99-106. Google Scholar

[35]

C. Timofte, Multiscale analysis of diffusion processes in composite media, Computers and Mathematics with Applications, 66 (2013), 1573-1580. doi: 10.1016/j.camwa.2012.12.003.  Google Scholar

[36]

C. Timofte, Multiscale modeling of heat transfer in composite materials, Romanian Journal of Physics, 58 (2013), 1418-1427. Google Scholar

[37]

C. Timofte, Multiscale analysis in nonlinear thermal diffusion problems in composite structures, Cent. Eur. J. Phys., 8 (2010), 555-561. doi: 10.2478/s11534-009-0141-6.  Google Scholar

show all references

References:
[1]

M. Amar, D. Andreucci and R. Gianni, Evolution and memory effects in the homogenization limit for electrical conduction in biological tissues, Math. Model. Methods Appl. Sci., 14 (2004), 1261-1295. doi: 10.1142/S0218202504003623.  Google Scholar

[2]

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, Differ. Integral Equations, 26 (2013), 885-912.  Google Scholar

[3]

J. L. Auriault, C. Boutin and C. Geindreau, Homogenization of Coupled Phenomena in Heterogenous Media, Wiley, 2010. doi: 10.1002/9780470612033.  Google Scholar

[4]

J. L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Int. J. of Heat and Mass Transfer, 37 (1994), 2885-2892. doi: 10.1016/0017-9310(94)90342-5.  Google Scholar

[5]

A. G. Belyaev, A. L. Pyatnitskiĭ and G. A. Chechkin, Averaging in a perforated domain with an oscillating third boundary condition, Sbornik: Mathematics, 192 (2001), 933-949. doi: 10.1070/SM2001v192n07ABEH000576.  Google Scholar

[6]

Y. Benveniste and T. Miloh, Imperfect soft and stiff interfaces in two-dimensional elasticity, Mech. Mater., 33 (2001), 309-323. doi: 10.1016/S0167-6636(01)00055-2.  Google Scholar

[7]

D. Brinkman, K. Fellner, P. Markowich and M. T. Wolfram, A drift-diffusion-reaction model for excitonic photovoltaic bilayers: Asymptotic analysis and a 2-D HDG finite element scheme, Math. Models Methods Appl. Sci., 23 (2013), 839-872. doi: 10.1142/S0218202512500625.  Google Scholar

[8]

R. Bunoiu and C. Timofte, On the homogenization of a two-permeability problem with flux jump,, work in progress., ().   Google Scholar

[9]

D. Cioranescu, A. Damlamian, P. Donato, G. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760. doi: 10.1137/100817942.  Google Scholar

[10]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620. doi: 10.1137/080713148.  Google Scholar

[11]

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

[12]

D. Cioranescu, P. Donato and R. Zaki, Asymptotic behavior of elliptic problems in perforated domains with nonlinear boundary conditions, Asymptot. Anal., 53 (2007), 209-235.  Google Scholar

[13]

I. Chourabi and P. Donato, Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptot. Anal., 92 (2015), 1-43.  Google Scholar

[14]

P. Donato, L. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: a memory effect, J. Math. Pures Appl., 87 (2007), 119-143. doi: 10.1016/j.matpur.2006.11.004.  Google Scholar

[15]

P. Donato and K. H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, Nonlinear Differ. Equ. Appl., 22 (2015), 1345-1380. doi: 10.1007/s00030-015-0325-2.  Google Scholar

[16]

P. Donato, K. H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect transmission problems, J. Math. Sci. (N. Y.), 176 (2011), 891-927. doi: 10.1007/s10958-011-0443-2.  Google Scholar

[17]

P. Donato and S. Monsurrò, Homogenization of two heat conductors with an interfacial contact resistance, Analysis and Applications, 2 (2004), 247-273. doi: 10.1142/S0219530504000345.  Google Scholar

[18]

P. Donato and I. Ţenţea, Homogenization of an elastic double-porosity medium with imperfect interface via the periodic unfolding method, Boundary Value Problems, 2013 (2013), 14pp.  Google Scholar

[19]

H. I. Ene and D. Poliševski, Model of diffusion in partially fissured media, Z. Angew. Math. Phys., 53 (2002), 1052-1059. doi: 10.1007/PL00013849.  Google Scholar

[20]

H. I. Ene and C. Timofte, Microstructure models for composites with imperfect interface via the periodic unfolding method, Asymptot. Anal., 89 (2014), 111-122.  Google Scholar

[21]

H. I. Ene, C. Timofte and I. Ţenţea, Homogenization of a thermoelasticity model for a composite with imperfect interface, Bull. Math. Soc. Sci. Math. Roumanie, 58 (2015), 147-160.  Google Scholar

[22]

H. I. Ene and C. Timofte, Homogenization results for a dynamic coupled thermoelasticity problem, Romanian Reports in Physics, 68 (2016), 979-989. Google Scholar

[23]

K. Fellner and V. Kovtunenko, A discontinuous Poisson-Boltzmann equation with interfacial transfer: Homogenisation and residual error estimate, Applicable Analysis, (2015), 1-22. doi: 10.1080/00036811.2015.1105962.  Google Scholar

[24]

M. Gahn, P. Knabner and M. Neuss-Radu, Homogenization of reaction-diffusion processes in a two-component porous medium with a nonlinear flux condition at the interface, and application to metabolic processes in cells, SIAM J. Appl. Math., 76 (2016), 1819-1843. doi: 10.1137/15M1018484.  Google Scholar

[25]

Z. Hashin, Thin interphase-imperfect interface in elasticity with application to coated fiber composites, Journal of the Mechanics and Physics of Solids, 50 (2002), 2509-2537. doi: 10.1016/S0022-5096(02)00050-9.  Google Scholar

[26]

H. K. Hummel, Homogenization for heat transfer in polycrystals with interfacial resistances, Appl. Anal., 75 (2000), 403-424. doi: 10.1080/00036810008840857.  Google Scholar

[27]

E. R. Ijioma, A. Muntean and T. Ogawa, Pattern formation in reverse smouldering combustion: A homogenization approach, Combustion Theory and Modelling, 17 (2013), 185-223. doi: 10.1080/13647830.2012.734860.  Google Scholar

[28]

E. C. Jose, Homogenization of a parabolic problem with an imperfect interface, Rev. Roum. Math. Pures Appl., 54 (2009), 189-222.  Google Scholar

[29]

A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, WIT-Press, Southampton, Boston, 2000. Google Scholar

[30]

K. H. Le Nguyen, Homogenization of heat transfer process in composite materials, JEPE, 1 (2015), 175-188.  Google Scholar

[31]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63.  Google Scholar

[32]

D. Polisevski, R. Schiltz-Bunoiu and A. Stanescu, Homogenization cases of heat transfer in structures with interfacial barriers, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 58 (2015), 463-473.  Google Scholar

[33]

D. Polisevski and R. Schiltz-Bunoiu, Heat conduction through a first-order jump interface, New Trends in Continuum Mechanics (M. Mihailescu-Suliciu ed.), Theta Series in Advanced Mathematics, 3 (2005), 225-230.  Google Scholar

[34]

D. Polisevski and R. Schiltz-Bunoiu, Diffusion in an intermediate model of fractured porous media, Bulletin Scientifique, Mathématiques et Informatique, 10 (2004), 99-106. Google Scholar

[35]

C. Timofte, Multiscale analysis of diffusion processes in composite media, Computers and Mathematics with Applications, 66 (2013), 1573-1580. doi: 10.1016/j.camwa.2012.12.003.  Google Scholar

[36]

C. Timofte, Multiscale modeling of heat transfer in composite materials, Romanian Journal of Physics, 58 (2013), 1418-1427. Google Scholar

[37]

C. Timofte, Multiscale analysis in nonlinear thermal diffusion problems in composite structures, Cent. Eur. J. Phys., 8 (2010), 555-561. doi: 10.2478/s11534-009-0141-6.  Google Scholar

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