On the homogenization of a two-conductivity problem with flux jump

Abstract

In this paper, we study the homogenization of a thermal diffusion problem in a highly heterogeneous medium formed by two constituents. The main characteristics of the medium are the discontinuity of the thermal conductivity over the domain as we go from one constituent to another and the presence of an imperfect interface between the two constituents, where both the temperature and the flux exhibit jumps. The limit problem, obtained via the periodic unfolding method, captures the influence of the jumps in the limit temperature field, in an additional source term, and in the correctors, as well.

Upscaling of a diffusion problem with interfacial flux jump leading to a modified Barenblatt model

“…The same effect was observed for a Neumann problem in [18], for a Robin problem in [17] and for a problem with flux jump in [14].…”

“…Indeed, there is a subtle interplay between the form and the scaling of the jump functions ℎ, , and . The solution 2 is related to 1 via the algebraic Equation 14 and depends, in addition, on ℎ.…”

“…Remark 3.5. If the function has more regularity, for instance ∈ 1 (Ω), then, according to (14), 2 itself belongs to 1 (Ω).…”

“…Proof The proof of the existence and uniqueness result is done by using the Lax‐Milgram theorem. The continuity of l can be proven by using the estimates obtained in [] and [] (see also [, Theorem 2.3]). We recall that the proof of the continuity and of the coercivity of the bilinear form a is known if (see []).…”

“…Such a flux jump can arise in relevant physical situations, as for instance when dealing with the electric double-layer effect in charged porous media (see [25] and the references therein). Several results were derived, by using homogenization techniques, for problems with flux jump, as for instance in [2,3,7,13,14,[25][26][27]30,31], and [15]. The case presented here is different from the ones studied in the papers cited above.…”