Transport properties of the Menger sponge

TL;DR

This paper investigates the transport properties of the Menger sponge, a fractal structure, by deriving its effective transport coefficient and comparing it to the Sierpinski sponge, providing insights into fractal systems.

Contribution

It introduces a method to derive the effective transport coefficient of the Menger sponge and compares its properties with the Sierpinski sponge, advancing understanding of fractal transport phenomena.

Findings

Derived the effective transport coefficient of the Menger sponge

Compared transport properties with the Sierpinski sponge

Provided insights into fractal and non-fractal domain transport

Abstract

The Menger sponge is a three-dimensional cube that comprises a self-similar, fractal domain and a non-fractal domain, both of which are continuous. Thus it is a useful heuristic model for natural and engineered fractal systems. For this purpose the effective transport coefficient associated with the transport properties (e.g., electrical conductivity, thermal conductivity) of the sponge is derived. Comparison is made to the Sierpinski sponge.