Heat transfer in a complex medium

TL;DR

This paper derives an effective heat transfer equation in a complex medium with many small particles, without assuming periodicity, enabling the design of materials for directed heat signal transmission.

Contribution

It introduces a new limiting equation for heat transfer in media with numerous small particles, relaxing periodicity assumptions and facilitating engineered heat signal pathways.

Findings

Derived the limiting heat equation for complex media with small particles.

Established conditions for particle size and distribution as they tend to zero.

Provided a foundation for designing materials with controlled heat signal transmission.

Abstract

The heat equation is considered in the complex medium consisting of many small bodies (particles) embedded in a given material. On the surfaces of the small bodies an impedance boundary condition is imposed. An equation for the limiting field is derived when the characteristic size $a$ of the small bodies tends to zero, their total number $\mathcal{N}(a)$ tends to infinity at a suitable rate, and the distance $d = d(a)$ between neighboring small bodies tends to zero: $a << d$, $\lim_{a\to 0}\frac{a}{d(a)}=0$. No periodicity is assumed about the distribution of the small bodies. These results are basic for a method of creating a medium in which heat signals are transmitted along a given line. The technical part for this method is based on an inverse problem of finding potential with prescribed eigenvalues.