Approach to steady state in the heat equation and the hyperbolic heat transfer equation

TL;DR

This paper compares the behavior of the classical heat equation and the hyperbolic heat transfer equation in modeling the approach to steady state, highlighting the hyperbolic model's physical relevance for small times.

Contribution

It demonstrates that the hyperbolic heat equation better captures the approach to steady state in finite systems, especially at small times, unlike the classical parabolic model.

Findings

Parabolic heat equation fails to describe steady state approach in infinite space.

Hyperbolic heat equation shows a finite time constant related to thermal relaxation.

Hyperbolic model is more physically accurate for small times in finite geometries.

Abstract

We investigate the spherically symmetric 1D ablation problem. We show that the parabolic heat equation fails to describe the approach to steady state in infinite space. The hyperbolic equation shows an approach to steady state with a time constant given by the thermal relaxation time. However the infinite geometry is rather unphysical and gives rise to a so-called zero mode. Therefore we also consider the finite problem with a large boundary at constant temperature. Then both equations show approach to steady state, but only the hyperbolic equation seems to be physically correct for small times.