Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations

Abstract

A dynamical nonequilibrium temperature has been proposed to describe relaxational equations for the heat\ud flux. This temperature provides an alternative description to the Maxwell-Cattaneo equation. In the linear\ud regime and in bulk systems both descriptions are equivalent but this is not so when nonlinear effects are\ud included. Here we explore the influence of nonlinear terms on the phase speed of heat waves in nonequilibrium\ud steady states in both theoretical models and we show that their predictions are different. This could allow to\ud explore which description is more suitable, when experiments on these situations will become available.\ud Furthermore, we have analyzed a nonlinear and nonlocal constitutive equation for the heat flux and we have\ud shown its analogy with the Navier-Stokes equation in the regime of phonon hydrodynamics in nanosystems.\ud This analogy allows one to define a dimensionless number for heat flow, analogous to the Reynolds number,\ud and to predict a critical heat flux where nonlinear effects could become dominant

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

“…(14) while the evolution of the heat flux is ruled by an equation of the type (16), then the use of a dynamical nonequilibrium temperature β [4,20,21,45], allows to obtain a nonlinear generalization of the classical Maxwell-Cattaneo-vernotte and Guyer-Krumhansl equations [5][6][7]. For instance, Eq.…”

“…Extensions of Eq. (1) to the nonlinear regime may be obtained in the framework of Extended Irreversible Thermodynamics (EIT) [6,7,21,27,28], a theory which upgrades the dissipative fluxes to the rank of independent thermodynamic variables [10,22,[29][30][31], while nonlinear generalizations of Eq. (2) have been obtained either in EIT [5], or in the framework of the thermomass theory [12][13][14], which also pays a special attention to nonlinear terms in the heat transport equation.…”

“…Three such approaches are the phonon hydrodynamics [5][6][7], the thermomass theory [12][13][14], and the dualphase-lag model [15,16]. All these models consider the heat carriers as a fluid, whose hydrodynamic-like equations of motion describe the heat transport.…”

“…One of them is the meaning of temperature and its relation with heat transport [3][4][5][6][7]. In fact, different degrees of freedom, far from equilibrium, may be associated with different "temperatures".…”

Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

“…This generalization leads to a hyperbolic transport equation instead of the parabolic one [2]. As an example, the non-local time constitutive equations for the heat flux and phonon hydrodynamics in nanosytems have been presented by Cimmelli, Sellitto and Jou [3]. 1 2 -order.…”