Series Solutions for Orthotropic Diffusion in a Cube

TL;DR

This paper derives explicit transient solutions for diffusion in a cube with orthotropic anisotropy, providing benchmark solutions for validating numerical models under various initial conditions.

Contribution

It presents the first explicit transient solutions for diffusion in a cube with orthotropic anisotropic conductivity, useful for validation and idealization of real systems.

Findings

Provides solutions for delta, Gaussian, step, and planar initial conditions.

Offers benchmarks for numerical code validation.

Addresses physically relevant anisotropic diffusion scenarios.

Abstract

Analytical solutions to heat or diffusion type equations are numerous, but there are rather few explicit solutions for conditions where the thermal conductivity or diffusion tensors are anisotropic. Such solutions have some use in making predictions for idealization of real systems, but are perhaps most useful for providing benchmark solutions which can be used to validate numerical codes. In this short paper, we present the transient solution to the diffusion equation in a cube under conditions of orthotropic anisotropy in the effective thermal conductivity or diffusion tensor. In particular, we consider the physically-relevant case of transport in a cube with no-flux boundaries for several initial conditions including: (1) a delta function, (2) a truncated Gaussian function, (3) a step function, and (4) a planar function. The potential relevance for each of these initial conditions in the context of validating numerical codes is discussed.