A Guide through Available Mixture Theories for Applications

In this article, we discuss the approaches used in the theory of mixtures and provide a critical review from a fresh perspective with the aim to identify limitations and open questions inherent to various versions of the theory. Such a discussion could serve as a guide to the formulation of a model for the problem at hand. We look into the frequently neglected area of kinematic description, drawing from this field criticism of third Truesdell's metaphysical principle. Balance laws are carefully formulated for partial quantities with discontinuities for both theory of interacting continua and for averaging theories. Further, we discuss definitions of mixture quantities together with Truesdell's metaphysical principles. Finally, we use classical irreversible thermodynamics to derive entropy production for mixture as a whole and thus reaching an extension of its classical form, evolution equations (phenomenological equations), and constitutive relations. As this fundamental step is very much dependent on the use of non equilibrium thermodynamics, we give a brief overview of the theories available.