Foundations of Mixture Theory

Abstract

We here present an axiomatic derivation of the field equations of continuum thermodynamics. In this derivation, in which we also employ the usual continuity assumptions of continuum theory, our cardinal precept is TRUESDELL’S metaphysical principle number 2, taken quite literally: not only each component body of the mixture but also each subbody of each component body and by extension each combination of subbodies of component bodies is required to obey the laws of thermodynamics. From this prospectus one may expect the occurrence of many fields not found in the classical theory and correspondingly more balance equations. However we show in the end how the derived fields and equations reduce to those of Lecture 5, leading in particular to two important observations about the classical theory. First, one should note that thepeculiar quantities of the classical theory should not be interpreted in the naive way suggested by the form in which they appear in the balance equations. For example, the partial heat flux h_a • n integrated over a surface cannot reasonably be interpreted as the heat flux into component a from the totality of components located on the opposite side of that surface, but as a more complicated average of effects occurring among elements on the same side and on opposite sides of the surface. Second, although we necessarily arrive at individual entropy inequalities, there is no rational argument that they should take the classical form.