The Sech(Xi)-type profiles: a Swiss-Army knife for exact analytical modelling of thermal diffusion and wave propagation in graded media

TL;DR

This paper introduces sech(xi)-type profiles as versatile, exact analytical solutions for modeling thermal diffusion and wave propagation in graded media, enabling flexible and precise analysis of heterogeneous materials.

Contribution

It extends the class of solvable profiles using Liouville and Darboux transformations, providing a quadrupole formulation for constructing complex effusivity profiles.

Findings

Quadrupole formulation enables building arbitrary complexity profiles.

Sech(xi)-type profiles provide exact solutions for both heat and wave propagation.

Profiles facilitate replacing staircase models with high-level analytical models.

Abstract

This work deals with exact analytical modelling of transfer phenomena in heterogeneous materials exhibiting one-dimensional continuous variations of their properties. Regarding heat transfer, it has recently been shown that by applying a Liouville transformation and multiple Darboux transformations, infinite sequences of solvable profiles of thermal effusivity can be constructed together with the associated temperature (exact) solutions, all in closed-form expressions (vs. the diffusion-time variable and with a growing number of parameters). In addition, a particular class of profiles, so-called sech(xi)-type profiles, exhibit high agility and in the same time parsimony. In this paper we go further into the description of these solvable profiles and their properties. Most importantly, their quadrupole formulation is provided which allows building smooth synthetic profiles of effusivity of arbitrary complexity and thereafter getting very easily the corresponding temperature dynamic response. Examples are given with increasing variability of effusivity and increasing number of elementary profiles. These highly flexible profiles are equally relevant for providing an exact analytical solution to wave propagation problems in 1D graded media (i.e. Maxwell's equations, acoustic equation, telegraph equation...). From now on, let it be for diffusion-like or wave-like problems, when the leading properties present (possibly piecewise-) continuously heterogeneous profiles, the classical staircase model can be advantageously replaced by a "high-level" quadrupole model consisting of one or more sech(xi)-type profiles, which makes the latter a true Swiss-Army knife for analytical modelling.