Measuring the insulating ability of anisotropic thermal conductors via principal Dirichlet eigenvalue

Abstract

We consider the thermal insulation property of homogeneous anisotropically heat-conducting bodies, i.e. those whose thermal tensor (matrix) A is constant throughout the body but not generally a constant times the identity. This anisotropy is a common feature of nano-composite materials. We propose using the principal Dirichlet eigenvalue λ of the associated elliptic differential operator −∇ ⋅ A∇ as a simple measurement for the insulating ability of the material, because the time scale of thermal flow is of the order of 1/λ. 1/λ is a generalization of the ‘R-value’ used in engineering practice as a measure of the insulating ability of isotropic conductors. If the thermal tensor A depends on parameters, e.g. inherited from nanostructure, then so does λ. It is important to know how λ depends on the parameters. For the material to be a good insulator, λ should be suppressed. But calculation or estimation of this principal elliptic eigenvalue, particularly over ranges of parameter values, is not a simple task. The focus of this paper is estimation – by fitted formulas and new exact bounds – of the principal elliptic Dirichlet eigenvalue of ellipses (2D) and ellipsoids (3D) using only simple expressions in the eigenvalues of the matrix A. Our simplest approximations and bounds avoid even the calculation of matrix eigenvalues and use in 2D merely the trace and determinant of A, and in 3D the trace and determinant of A and the trace of A2. The new bounds are shown to imply an extremal property in homogenization theory and a new condition for enhancement in Taylor dispersion. Recently, Zheng, Forest, Lipton, Zhou and Wang published formulas for the thermal tensor A in the case of strong extensional fiber-type flow of a polymer with dilute rod-like nano-inclusions, including explicitly the influence of the probability distribution of inclusion orientation. Our results are combined with these formulas to quantify the effect of variations of the probability distribution on the insulating ability of the composite.