Heat conduction in composite media involving imperfect contact and perfectly conductive inclusions

Abstract

We study the thermal properties of a composite material made up of a medium hosting an ε-periodic array of perfect thermal conductors. The thermal potentials of the two phases are coupled across the interface through a non-standard imperfect contact transmission condition, involving the external thermal flux and a proportionality coefficient D 0 ε α , where α ∈ R is a scaling parameter and D 0 > 0 accounts for the imperfect contact. We perform the homogenization for all the scalings α ∈ R and we compare the resulting models with the perfect contact transmission case addressed in [6,7], letting D 0 → 0, where this is meaningful.

Asymptotic analysis for non-local problems in composites with different imperfect contact conditions

“…We point out that one can consider also other scalings with respect to ε in (2.6) and (2.7) (see [11]), leading to different macroscopic problems; however, we decided to treat here only the present case, which seems to be the most interesting one, as pointed out in the Introduction. We also remark that our homogenization results can be obtained, without additional difficulties, even for a diffusion matrix κ = κ(x, y), Y -periodic in the second variable and sufficiently smooth in the first one.…”

“…For the mathematical aspects behind the behaviour of these kinds of materials, we refer to [2,14,15,16,17,22,24,28,29,31,32,33], where such problems are studied in various contexts like heat diffusion, electric conduction, petroleum engineering industry, wave equations or elastic properties of perforated materials. A crucial feature of our and similar models is represented by the fact that, while the temperature u out ε in the hosting medium satisfies a standard heat equation, inside each inclusion the temperature u int ε depends only on time and is governed by an ordinary differential equation, involving a non-standard condition of non-local type, in which the time-variation of the temperature of the inner phase is determined by the global thermal flux coming from the outer phase (see, for instance, [6,7,11]). However, differently from the previous contributions, where only one type of inclusions was considered, in the present paper we address the case in which we have two types of fillers, having dissimilar thermal features.…”

“…Actually, the models treated in the references quoted at the beginning of the Introduction deal with perfect contact conditions which can be seen as an approximation of the more realistic imperfect contact conditions. The imperfect contact conditions we propose here involve the microscopic geometry of the problem through the characteristic length ε of the inclusions and two amplitude factors, similarly as described in [11], where a hierarchy of possible scalings in the period of the microscopic geometry is considered. As a consequence of the investigation in [11], only two scalings result to be critical, in the sense that they preserve memory of the amplitude factor in the macroscopic model, obtained after passing to the limit ε → 0.…”

“…The imperfect contact conditions we propose here involve the microscopic geometry of the problem through the characteristic length ε of the inclusions and two amplitude factors, similarly as described in [11], where a hierarchy of possible scalings in the period of the microscopic geometry is considered. As a consequence of the investigation in [11], only two scalings result to be critical, in the sense that they preserve memory of the amplitude factor in the macroscopic model, obtained after passing to the limit ε → 0. It is worthwhile to notice that hierarchies of scalings are often considered in homogenization problems (see, for instance, [5,13,16,25,29,36] and the references therein).…”

“…The other goal of the paper is to perform the limit in D 1 and D 2 . We consider not only the case where D 1 and/or D 2 go to 0, for which the perfect contact condition is recovered (see [6,11]), but also the case where D 1 and/or D 2 go to +∞, which accounts for insulated materials. The limits in ε and in D 1 , D 2 are considered in different orders, with the aim of comparing the final results and identifying the cases in which the two limits commute.…”