### abstract

- The paper considers developments of constitutive theories in Eulerian description for compressible as well as incompressible ordered homogeneous and isotropic thermofluids in which the deviatoric Cauchy stress tensor and the heat vector are functions of density, temperature, temperature gradient, and the convected time derivatives of the strain tensors of up to a desired order. The fluids described by these constitutive theories are called ordered thermofluids due to the fact that the constitutive theories for the deviatoric Cauchy stress tensor and heat vector are dependent on the convected time derivatives of the strain tensor up to a desired order, the highest order of the convected time derivative of the strain tensor in the argument tensors defines the 'order of the fluid'. The admissibility requirement necessitates that the constitutive theories for the stress tensor and heat vector satisfy conservation laws, hence, in addition to conservation of mass, balance of momenta, and conservation of energy, the second law of thermodynamics, that is, Clausius-Duhem inequality must also be satisfied by the constitutive theories or be used in their derivations. If we decompose the total Cauchy stress tensor into equilibrium and deviatoric components, then Clausius-Duhem inequality and Helmholtz free energy density can be used to determine the equilibrium stress in terms of thermodynamic pressure for compressible fluids and in terms of mechanical pressure for incompressible fluids, but the second law of thermodynamics provides no mechanism for deriving the constitutive theories for the deviatoric Cauchy stress tensor. In the development of the constitutive theories in Eulerian description, the covariant and contravariant convected coordinate systems, and Jaumann measures are natural choices. Furthermore, the mathematical models for fluids require Eulerian description in which material point displacements are not measurable. This precludes the use of displacement gradients, that is, strain measures, in the development of the constitutive theories. It is shown that compatible conjugate pairs of convected time derivatives of the deviatoric Cauchy stress and strain measures in co-, contravariant, and Jaumann bases in conjunction with the theory of generators and invariants provide a general mathematical framework for the development of constitutive theories for ordered thermofluids in Eulerian description. This framework has a foundation based on the basic principles and axioms of continuum mechanics but the resulting constitutive theories for the deviatoric Cauchy stress tensor must satisfy the condition of positive work expanded, a requirement resulting from the entropy inequality. The paper presents a general theory of constitutive equations for ordered thermofluids which is then specialized, assuming first-order thermofluids, to obtain the commonly used constitutive theories for compressible and incompressible generalized Newtonian and Newtonian fluids. It is demonstrated that the constitutive theories for ordered thermofluids of all orders are indeed rate constitutive theories. We have intentionally used the term 'thermofluids' as opposed to 'thermoviscous fluids' due to the fact that the constitutive theories presented here describe a broader group of fluids than Newtonian and generalized Newtonian fluids that are commonly referred as thermoviscous fluids. © 2012 Springer-Verlag.