Variational Formulation of Linear Equations of Coupled Thermohydrodynamics and Heat Conductivity
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Variational Formulation of Linear Equations of Coupled Thermohydrodynamics and Heat Conductivity P. A. Belov1* , S. A. Lurie1, 2, 3** , and V. N. Dobryanskiy2*** (Submitted by A. V. Lapin) 1
Institute of Applied Mechanics of Russian Academy of Sciences, Moscow, 125040 Russia 2 Moscow Aviation Institute (National Research University), Moscow, 125993 Russia 3 Federal Research Center “Computer Sciences and Control,” Russian Academy of Sciences, Moscow, 119333 Russia Received April 1, 2020; revised April 10, 2020; accepted April 18, 2020
Abstract—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. DOI: 10.1134/S1995080220100042 Keywords and phrases: irreversible processes, non-integrable variational forms, thermoelasticity, thermohydrodynamics, Fourier’s thermal conductivity, hyperbolic thermal conductivity.
1. INTRODUCTION The study of the problems of thermodynamics of media, taking into account the irreversibility and connectivity of the processes of mechanical deformation and heat transfer, is extremely in demand both in connection with the intensive development of technological processes and in connection with the need to predict the behavior of materials under conditions of intense physical fields. Nevertheless, these problems appear to be very complex and not fully resolved to date. For example, it is known that even the classical theory of heat transfer becomes insufficient to describe the distribution of heat at low temperatures [1], as well as to understand the thermoelastic properties of small-sized systems [2, 3], for modeling nanostructured microelectronic components and various thermoelectric devices [4]. Similar problems of heat and mass transfer require the involvement of more general Maxwell– Cattaneo type models [5, 6], which are free from the contradictions of diffusion type models. This component
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