Faculty of Physics - University of Bucharest

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FACULTY OF PHYSICS

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Conference: Bucharest University Faculty of Physics 2006 Meeting

Section: Quantum Mechanics and Statistical Physics Seminar

Title:

Upscaling in Nonlinear Thermal Diffusion Problems in Composite Materials

Authors:

Claudia Timofte

Affiliation:

Department of Mathematics, Faculty of Physics, University of

Bucharest, P.O.Box MG-11, Bucharest-Magurele, Romania.

E-mail

claudiatimofte@yahoo.com

Keywords:

Heat transfer, interfacial thermal barrier, homogenization.

Abstract:

This talk deals with the homogenization of a nonlinear problem arising in the modelling of thermal diffusion in a two-component composite with an interfacial thermal barrier. We consider, at the microscale, a periodic structure formed by two connected components representing two materials with different thermal features. We assume that at the interface between our two materials the flux is continuous and depends in a nonlinear way on the jump of the temperature field. Since the characteristic sizes of these two components are small compared with the macroscopic length-scale of the flow domain, we can apply an homogenization procedure. As usual in homogenization, we shall be interested in obtaining a suitable description of the asymptotic behavior, as the small parameter which characterizes the sizes of our two regions tends to zero, of the temperature field in the periodic composite. The asymptotic behavior of the solution of such a problem will be governed by a new elliptic boundary-value problem with an extra zero-order term that captures the effect of the interfacial barrier. The homogenized problem will describe, in fact, the effective thermal conductivity of the composite. The approach we use is the so-called energy method introduced by L. Tartar for studying homogenization problems. It consists of constructing suitable test functions that are used in our variational problems. However, it is worth mentioning that another possible way to get the homogenized problem could be to use the two-scale convergence technique.