Unified formulas for the effective conductivity of fibrous composites with circular inclusions and parallelogram periodicity and its influence on thermal gain in nanofluids

TL;DR

This paper develops unified analytical formulas for the effective conductivity of fibrous composites with circular inclusions in parallelogram cells, and explores their impact on thermal enhancement in nanofluids.

Contribution

It introduces a systematic method combining asymptotic homogenization and elliptic functions to derive formulas for any parallelogram cell, including two-phase fibrous composites with various interface conditions.

Findings

Derived explicit formulas for effective conductivity coefficients.

Validated the model with numerical examples and comparisons.

Provided an accessible algorithm and software for practical calculations.

Abstract

A two-dimensional three-phase conducting composite with coated circular inclusions, periodically distributed in a parallelogram, is studied. The phases are assumed to be isotropic, and perfect contact conditions at the interfaces are considered. The effective behavior is determined by combining the asymptotic homogenization method with elements of the analytic function theory. The solution to local problems is sought as a series of Weierstrass elliptic functions and their derivatives with complex undetermined coefficients. The effective coefficients depend on the residue of such a solution, which in turn depends on products of vectors and matrices of infinite order. Systematic truncation of these vectors and matrices provides unified analytical formulas for the effective coefficients for any parallelogram periodic cell. The corresponding formulas for the particular cases of two-phase fibrous composites with perfect and imperfect contact at the interface are also explicitly provided. The results were applied to derive the critical normalized interfacial layer thickness and to analyze the enhancement of thermal conductivity in fibrous composites with annular cross sections. Furthermore, using a reiterated homogenization method, the analytical approach allows us to study the gains in the effective thermal conductivity tensor with thermal barriers and parallelogram cells. Numerical examples and comparisons validate the model. A simple and validated algorithm is provided that allows the calculation of effective coefficients for any parallelogram, any truncation order, and high fiber volume fractions very close to percolation. The programs created for validation are available in a freely accessible repository.