ON THERMODYNAMICS AND THE NATURE OF THE SECOND LAW FOR MIXTURES OF INTERACTING CONTINUA

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Abstract
This paper, which may be regarded as a continuation of previous papers (9, 10) on thermodynamics of single phase continue, is concerned with a new approach to thermomechanics of multiphase continua and extends the previous ideas and procedure to mixtures of interacting continua. In particular, it contains (a) a proposal of a new approach for obtaining restrictions on constitutive equations, (b) an appropriate mathematical statement of the second law for mixtures and (c) the nature of restrictions placed by the latter on constitutive results representing the thermomechanical behavior of mixtures with different constituent temperatures. Our point of departure is the introduction of balances of entropy and the use of a single energy equation for the whole mixture as an identity for all motions and all temperature distributions after the elimination of the external fields. This procedure is in contrast with the existing approach in most of the current literature on continuum theories of mixtures based on the use of a Clausius-Duhem type inequality (or similar entropy inequalities) for mixtures. Our interpretation of the second law is similar to that of the previous paper and leads us to postulate an inequality which reflects the fact that for every process associated with a dissipative mixture, a part of the external mechanical work is always converted into heat and this cannot be withdrawn from the mixture as mechanical work. The restriction on the heat conduction vectors is considered separately and is confined to equilibrium cases in which heat flow is steady. Also, a restriction on the specific internal energies is derived when the mixture is in the state of mechanical equilibrium with all its constituents at a common spatially homogeneous temperature. The remainder of the paper deals with the constitutive results and thermodynamic restrictions for inviscid fluids, a fairly detailed consideration of the problem of an incompressible Newtonian viscous fluid flowing through a rigid porous solid whose temperature may be different from that of the fluid, as well as some additional remarks on the implication of the use of an entropy inequality of the Clausius-Duhem type for mixtures as contrasted with the thermodynamic restrictions resulting from the procedure proposed here and from our present interpreation of the second law.

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