Symmetry Properties of Models for Reversible and Irreversible Thermodynamic Processes

Abstract

The problem of formulating variational models for irreversible processes of media deformation is considered in this paper. For reversible processes, the introduction of variational models actually comes down to defining functionals with a given list of arguments of various tensor dimensions. For irreversible processes, an algorithm based on the principle of stationarity of the functional is incorrect. In this paper, to formulate a variational model of irreversible deformation processes with an expanded range of coupled effects, an approach is developed based on the idea of the introduction of the non-integrable variational forms that clearly separate dissipative processes from reversible deformation processes. The fundamental nature of the properties of symmetry and anti-symmetry of tensors of physical properties in relation to multi-indices characterizing independent arguments of bilinear forms in the variational formulation of models of thermomechanical processes has been established. For reversible processes, physical property tensors must necessarily be symmetric with respect to multi-indices. On the contrary, for irreversible thermomechanical processes, the tensors of physical properties that determine non-integrable variational forms must be antisymmetric with respect to the permutation of multi-indices. As a result, an algorithm for obtaining variational models of dissipative irreversible processes is proposed. This algorithm is based on determining the required number of dissipative channels and adding them to the known model of a reversible process. Dissipation channels are introduced as non-integrable variational forms that are linear in the variations of the arguments. The hydrodynamic models of Darcy, Navier–Stokes, and Brinkman are considered, each of which is determined by a different set of dissipation channels. As another example, a variational model of heat transfer processes is presented. The equations of heat conduction laws are obtained as compatibility equations by excluding the introduced thermal potential from the constitutive equations for temperature and heat flux. The Fourier and Maxwell–Cattaneo equations and the generalized heat conduction laws of Gaer–Krumhansl and Jeffrey are formulated.