On the thermodynamics of rate-type fluids and phase transitions. I. Rate-type fluids

Abstract A Maxwell's rate-type fluid without shear stresses is considered. A constitutive equation relating linearly the rate of pressure p to the rate of density ϱ and the rate of absolute temperature T at any fixed state ( ϱ , T , p ), is postulated. It is also assumed that for homogeneous processes the rate of heat and the rate of work at any state ( ϱ , T , p ) are given differential forms of the rates of density and temperature. The two laws of thermodynamics are stated in a very classical way, i.e., the existence of the thermodynamic potentials, internal energy U and entropy S as functions of state ( ϱ , T , p ) is required such that certain relations between their rates and the rates of heat and work are verified. Two over determined systems of partial differential equations and additional equality/inequalities necessary for the existence of the above potentials are deduced. The restrictions imposed by the two laws on the constitutive functions is discussed for fast processes when the relation p − ϱ − T is path dependent or path independent. A Clapeyron type formula is found as a necessary and sufficient condition for the existence of the instantaneous elastic (path independent) response. The restrictions of the two laws on the constitutive functions at the equilibrium states are also given. The construction of U , S as solutions of the over determined systems is explicitly given. It is proved that ψ=U − TS is the free energy function of the thermodynamic system if the dynamic pressure–density modulus is greater than the equilibrium one. An energy identity/inequality expressed in terms of the availability is also deduced as a consequence of the two laws of thermodynamics and the other balance laws which govern the motion of the fluid. These general results are further specialized in Part II to obtain van der Waals rate-type constitutive equations able to describe phase transition in fluids.