Thermodynamic derivation of the heat conduction equation and the dynamic boundary value problem for thermoelastic materials and fluids

Abstract

It is shown that the dynamic boundary value problem and the heat conduction equation for some simple materials are derivable from the first and second laws of thermodynamics. The dynamic boundary value problem, the heat conduction equation and two variational principles are derived for thermoelastic materials with time-dependent properties, for the case when the volume and surface forces are not “dead”, and when the free energy of the material depends upon the temperature. It is also shown that the conventional form of the heat conduction equation for geometrically nonlinear anisotropic elastic media does not satisfy the principle of material frame indifference. A new form of the heat conduction equation is offered. The heat conduction equation for the Navier-Stokes fluid and the dynamic boundary value problem for an elastic fluid are obtained. The elastic fluid is proved to be the only simple fluid without “memory”.