Variational Formulation of Linear Equations of Coupled Thermohydrodynamics and Heat Conductivity

We consider the coupled processes of thermohydrodynamics and heat conduction and construct a variational model of such coupled problems using four-dimensional space-time continuum where time is an equal coordinate along with spatial coordinates. In this consideration 3D subspace of generalized four-dimensional pseudo-continuum is associated with three-dimension deformed media. In the general case we consider irreversible processes and use the variational principle of possible displacements and the variational Sedov’s principle to construct variation model. It is assumed that the 4D pseudo-continuum considered below is transversely isotropic in the direction of the unit vector of time. Thus, an asymmetric 4D stress tensor preserves the symmetry properties with respect to spatial tangential stresses in 3D subspaces. It is shown that the proposed version of the model allows us to formulate the full range of consistent thermomechanical and thermodynamic physical relations, and the system of governing equations includes, as special cases, the equations of thermoelasticity, thermohydrodynamics, the linear Navier–Stokes equations for compressible and incompressible media, the equations of heat balance with the laws of thermal conductivity of Fourier, Maxwell–Cattaneo.