Derivation of a homogenized two-temperature model from the heat equation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1583-1613.

This work studies the heat equation in a two-phase material with spherical inclusions. Under some appropriate scaling on the size, volume fraction and heat capacity of the inclusions, we derive a coupled system of partial differential equations governing the evolution of the temperature of each phase at a macroscopic level of description. The coupling terms describing the exchange of heat between the phases are obtained by using homogenization techniques originating from [D. Cioranescu, F. Murat, Collège de France Seminar, vol. II. Paris 1979-1980; vol. 60 of Res. Notes Math. Pitman, Boston, London (1982) 98-138].

DOI : 10.1051/m2an/2014011
Classification : 35K05, 35B27, 76T05, 35Q79, 76M50
Mots-clés : heat equation, homogenization, infinite diffusion limit, thermal nonequilibrium models 
@article{M2AN_2014__48_6_1583_0,
     author = {Desvillettes, Laurent and Golse, Fran\c{c}ois and Ricci, Valeria},
     title = {Derivation of a homogenized two-temperature model from the heat equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1583--1613},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {6},
     year = {2014},
     doi = {10.1051/m2an/2014011},
     mrnumber = {3264366},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2014011/}
}
TY  - JOUR
AU  - Desvillettes, Laurent
AU  - Golse, François
AU  - Ricci, Valeria
TI  - Derivation of a homogenized two-temperature model from the heat equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2014
SP  - 1583
EP  - 1613
VL  - 48
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2014011/
DO  - 10.1051/m2an/2014011
LA  - en
ID  - M2AN_2014__48_6_1583_0
ER  - 
%0 Journal Article
%A Desvillettes, Laurent
%A Golse, François
%A Ricci, Valeria
%T Derivation of a homogenized two-temperature model from the heat equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2014
%P 1583-1613
%V 48
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2014011/
%R 10.1051/m2an/2014011
%G en
%F M2AN_2014__48_6_1583_0
Desvillettes, Laurent; Golse, François; Ricci, Valeria. Derivation of a homogenized two-temperature model from the heat equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 6, pp. 1583-1613. doi : 10.1051/m2an/2014011. http://www.numdam.org/articles/10.1051/m2an/2014011/

[1] G. Bal, Transport through diffusive and nondiffusive regions, embedded objects and clear layers. SIAM J. Appl. Math. 62 (2002) 1677-1697. | MR | Zbl

[2] M. Bellieud, Homogenization of evolution problems for a composite medium with very small and heavy inclusions. ESAIM: COCV 11 (2005) 266-284. | Numdam | MR | Zbl

[3] M. Bellieud, A notion of capacity related to elasticity. Applications to homogenization. Arch. Rational Mech. Anal. 203 (2012) 137-187. | MR | Zbl

[4] M. Bellieud, C. Licht and S. Orankitjaroen, Nonlinear capacitary problems for a non periodic distribution of fibers. Appl. Math. Res. Express 2014 (2014) 1-51. | MR | Zbl

[5] F. Boyer and P. Fabrie, Éléments d'analyse pour l'étude de quelques modèles d'écoulements de fluides visqueux incompressibles. Math. Appl., vol. 52. Springer Verlag, Berlin, Heidelberg (2006). | MR | Zbl

[6] C.E. Brennen, Fundamentals of Multiphase Flows. Cambridge University Press (2005). | Zbl

[7] H. Brezis, Analyse Fonctionnelle. Théorie et Applications. Masson, Paris (1987). | MR | Zbl

[8] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs. In Nonlinear Partial Differential Equations and their Applications, Collège de France Seminar vol. II, Paris 1979-1980; vol. 60 of Res. Notes Math. Pitman, Boston, London (1982) 98-138. | MR | Zbl

[9] L. Desvillettes, F. Golse and V. Ricci, The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow. J. Stat. Phys. 131 (2008) 941-967. | MR | Zbl

[10] F. Fichot, F. Duval, N. Trégourès, C. Béchaud and M. Quintard, The impact of thermal non-equilibrium and large-scale 2D/3D effects on debris bed reflooding and coolability. Nucl. Eng. Design 236 (2006) 2144-2163.

[11] V.A. L'Vov and E. Ya. Hruslov, Perturbations of a viscous incompressible fluid by small particles. Theor. Appl. Quest. Differ. Equ. Algebra 267 (1978) 173-177. | Zbl

[12] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, vol. 1. Dunod, Paris (1968). | Zbl

[13] A. Mikelic and M. Primicerio, Homogenization of heat conduction in materials with periodic inclusions of a perfect conductor. In Progress in partial differential equations: calculus of variations, applications. Pont-Mousson, 1991, vol. 267 of Pitman Res. Notes Math. Ser. Longman Sci. Tech., Harlow (1992) 244-256. | MR | Zbl

[14] A. Mikelic, M. Primicerio, Homogenization of the heat equation for a domain with a network of pipes with a well-mixed fluid. Ann. Mat. Pura Appl. 166 (1994) 227-251. | MR | Zbl

[15] F. Petit, F. Fichot, M. Quintard, Ecoulement diphasique en milieu poreux: modèle à non-équilibre local. Int. J. Therm. Sci. 38 (1999) 239-249. | Zbl

[16] P.-A. Raviart and J.-M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Masson, Paris, 1983. | MR | Zbl

[17] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR | Zbl

Cité par Sources :