Thermal pulse propagation beyond the Maxwell–Cattaneo theory: Application to one-dimensional nanosystems

Abstract

A non-local and non-linear thermodynamical model of heat transfer at nanoscale beyond the well-known Maxwell–Cattaneo theory is derived. The compatibility of the proposed model with second law of thermodynamics is proved. The model is subsequently used to investigate the propagation of a heat pulse in one-dimensional nanosystems in the linear case. The predicted results are compared with those arising from the Maxwell–Cattaneo theory in order to point out the possible influence both of the non-local effects, and of the relaxation effects of the higher-order fluxes. Some problems related to initial data and boundary conditions are also discussed.

Some decay properties in extensible thermoelastic beam with Gurtin–Pipkin’s law

“…Following Di Domenico et al [18] and Giorgi and Naso [7], the heat equation can be deduced directly from the entropy balance equation using the Gibbs relation [24] ρ(E t − T S t ) − P = 0, where E is the internal energy density per unit mass and P = i,j σ ij ε ijt . In fact, this relation allows us to rewrite the internal energy balance equation, ρE t − P + q x = 0.…”

“…The studied systems are obtained by replacing Fourier's law by Gurtin-Pipkin's law without any thermodynamic consideration or rigorous demonstration. For this, the studied systems in the works by Dridi [25] and Liu et al [26] are different from equation (18).…”

“…Following Di Domenico et al [18] and Giorgi and Naso [7], the heat equation can be deduced directly from the entropy balance equation using the Gibbs relation [24]…”

“…The constitutive equations for thermal conduction are introduced according to the theory of thermoelastic plates due to Chadwick [17]. In fact, Domenico et al [18] used the Gibbs relation in the context of Cattaneo's law in the frame of extended irreversible thermodynamics according to Jou et al [19] and Lebon et al [20]. Following Di Domenico et al [18], Giorgi and Naso [7], the heat equation in the context of Cattaneo's law is introduced using the Gibbs relation, which turns to Gurtin-Pipkin's law under a suitable choice of the memory kernel.…”

Constitutive Modeling with Single and Dual Internal Variables

“…where τ denotes a relaxation time and a is a material parameter. Despite the use of numerous alternative models and extensive research, the question of a suitable non-Fourier heat conduction model still remains open [32,33].…”