On the M\"uller paradox for thermal-incompressible media

TL;DR

This paper addresses the M"uller paradox by redefining incompressibility as a limit case, showing that the entropy principle holds at pressures below a critical value, thus resolving the paradox in thermodynamics.

Contribution

It introduces a new definition of incompressibility as a limiting case of quasi thermal-incompressible media, ensuring the entropy principle's validity at certain pressure ranges.

Findings

Entropy principle holds for pressures below a critical threshold.

Perfect incompressibility is impossible at very high pressures.

The approach applies to both fluids and hyperelastic media.

Abstract

In his monograph Thermodynamics, I. M\"uller proves that for incompressible media the volume does not change with the temperature. This M\"uller paradox yields an incompatibility between experimental evidence and the entropy principle. This result has generated much debate within the mathematical and thermodynamical communities as to the basis of Boussinesq approximation in fluid dynamics. The aim of this paper is to prove that for an appropriate definition of incompressibility, as a limiting case of quasi thermal-incompressible body, the entropy principle holds for pressures smaller than a critical pressure value. The main consequence of our result is the physically obvious one, that for very large pressures, no body can be perfectly incompressible. The result is first established in the fluid case. In the case of hyperelastic media subject to large deformations the approach is similar, but with a suitable definition of the pressure associated with convenient stress tensor decomposition.