Variational formulation of gradient irreversible thermodynamics

This work proposes the elaboration of the variational principle of L.I. Sedov for modeling dissipative processes. The formulated variational principle makes it possible to propose the dissipative models using the known model of a reversible process (the known Lagrangian), adding the required number of dissipation channels. Dissipation channels are non-integrable variational forms that are linear in the variations of the arguments. The arguments of the dissipation channels are the generalized variables of the corresponding bilinear terms of the Lagrangian. Variational models of heat transfer processes are considered as examples. The paper introduces the thermal potential, which is taken as the main kinematic variable. The temperature and heat flux are determined from the expression of the possible work done on variations of the first derivatives of the thermal potential, by analogy with continuum mechanics, where internal forces do the possible work on the strain variations. The equations of heat conduction laws of the considered heat transfer models are obtained as compatibility equations by eliminating the thermal potential from the equations of constitutive relations for temperature and heat flow. It is shown that the proposed procedure for elaboration of the dissipative models makes it possible to obtain the laws of thermal conductivity of Fourier, Maxwell - Cattaneo, Gaer - Krumhaksl, Jeffrey and more general laws of thermal conductivity. For the simplest heat transfer model, a single dissipation channel was introduced, which made it possible to obtain a heat transfer equation containing the second and first time derivatives. This model takes into account the wave properties and dissipation by the diffusion mechanism. In a particular case, it is reduced to the classical model of heat conduction. For more general gradient models of heat transfer, additional dissipation channels are sequentially introduced. In accordance with the differential order of the heat balance equation, the variational method makes it possible to formulate a consistent spectrum of boundary conditions at each non-singular point of the surface. In addition, for a boundary value problem in time, the variational principle determines pairs of alternative conditions at the initial and final times of the process under consideration.