Series expansion for the effective conductivity of a periodic dilute composite with thermal resistance at the two-phase interface

Article type: Research Article

Authors: Dalla Riva, Matteo^{a; b} | Musolino, Paolo^{c; *} | Pukhtaievych, Roman^c

Affiliations: [a] Department of Mathematics, The University of Tulsa, 800 South Tucker Drive, Tulsa, Oklahoma 74104, USA. E-mail: [email protected] | [b] Department of Mathematics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK | [c] Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, Padova 35121, Italy. E-mails: [email protected], [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We study the asymptotic behavior of the effective thermal conductivity of a periodic two-phase dilute composite obtained by introducing into an infinite homogeneous matrix a periodic set of inclusions of a different material, each of them of size proportional to a positive parameter ϵ. We assume that the normal component of the heat flux is continuous at the two-phase interface, while we impose that the temperature field displays a jump proportional to the normal heat flux. For ϵ small, we prove that the effective conductivity can be represented as a convergent power series in ϵ and we determine the coefficients in terms of the solutions of explicit systems of integral equations.

Keywords: Effective conductivity, periodic dilute composite, singularly perturbed domain, asymptotic expansion, non-ideal contact condition

DOI: 10.3233/ASY-181495

Journal: Asymptotic Analysis, vol. 111, no. 3-4, pp. 217-250, 2019