#### Vol. 3, No. 6, 2008

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Variational principles for heat conduction in dissipative continua

### Stanisław Sieniutycz

Vol. 3 (2008), No. 6, 1135–1149
##### Abstract

Applying some results of nonequilibrium statistical mechanics obtained in the framework of Grad’s theory we evaluate nonequilibrium corrections $\Delta s$ to the entropy $s$ of resting incompressible continua in terms of the nonequilibrium density distribution function, $f$. To find corrections $\Delta e$ to the energy $e$ or kinetic potential $L$ we apply a relationship that links energy and entropy representations of thermodynamics. We also evaluate the coefficients of the wave model of heat conduction, such as relaxation time, propagation speed, and thermal inertia. With corrections to $L$ we then formulate a quadratic Lagrangian and a variational principle of Hamilton’s (least action) type for a fluid with heat flux, or other random-type effect, in the field or Eulerian representation of the fluid motion. Results that are significant in the hydrodynamics of real incompressible fluids at rest and their practical applications are discussed. In particular, we discuss canonical and generalized conservation laws and show the satisfaction of the second law of thermodynamics under the constraint of canonical conservation laws. We also show the significance of thermal inertia and so-called thermal momentum in the variational formulation.

##### Keywords
wave equations, variational principles, thermal inertia, conservation laws