On the Müller paradox for thermal-incompressible media

In his monograph Thermodynamics, I. Müller proves that for incompressible media the volume does not change with the temperature. This Müller 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 case of hyperelastic media subject to large deformations, the approach is similar, but with a suitable definition of the pressure associated with a convenient stress tensor decomposition.