On high-order approximations for describing the lagging behavior of heat conduction

Abstract

This paper is dedicated to high-order effects of thermal lagging in correlation with heat transfer models in micro- or nanoscale, relating the number of energy carriers and the associated resonance phenomenon under high-frequency excitations. Thus, a class of constitutive equations is considered for the heat flux describing high-order effects in the lagging behavior of heat transport. Tzou’s model, which is based on time-differential dual-phase-lag approximations of heat conduction, is generalized, incorporating the microstructural interaction effect in the fast-transient process of heat transport. More precisely, polynomial approximations of order n for the heat flux vector and of order m for the gradient of the temperature variation are considered. Further, well-posedness is established for solutions of the specific initial boundary value problems for the mathematical model when: (i) [Formula: see text]; (ii) [Formula: see text]; and (iii) [Formula: see text]. This means that uniqueness and continuous dependences of solutions are established with respect to the given prescribed data. With this aim, some modified initial boundary value problems related to the operators of the constitutive equation are introduced and suitable time-weighted measures of the solution are presented, and are used to establish the uniqueness theorems and to generate some estimates describing the continuous dependence of solutions with respect to the given data for each of the three cases specified. Moreover, the spatial behavior of the transient solutions is studied and an influence domain result is established for [Formula: see text], while for [Formula: see text] some exponential decay estimates of Saint-Venant type are established. All results are established without restrictions on the characteristic parameters of the constitutive equation other than that the product of the coefficients of the time derivatives of greatest order of the two operators involved in the constitutive equation is positive.

A generalized heat conduction model of higher-order time derivatives and three-phase-lags for non-simple thermoelastic materials

“…Quintanilla and Racke 44 observed that there an area of influence result when , however when they have founded some spatial approximations explaining the Phragmén–Lindelöf alternative. It is worth noting that our choices of the HOTD parameters are capable to involve several models of heat conduction: when we take , we have a diffusive behavior; but by taking , we get a wavelike behavior 6 , 30 .…”

“…In one of these efforts, Abouelregal 27 – 29 introduced some generalized models of heat conduction including higher-order time-derivative. Earlier, Chiriţă 30 investigated the high-order Fourier law to illustrate the lagging performance of heat transfer.…”

Spatial Behavior of the Steady State Vibrations in a Dual-Phase-Lag Rigid Conductor

“…On the other side the spatial behavior of the transient solutions, for the model in concern, was established by Chiriţă [5]. It was shown there that (a) for n = m + 1 there was established an influence domain result (the corresponding constitutive equation describes the wave behavior), while (b) for n = m there was established some exponential decay estimates of the Saint-Venant type (the corresponding constitutive equation describes the diffusive behavior), without any restriction on the coefficients of the constitutive equation (2) other than a n b m > 0 .…”

“…The way in which the heat-affected zone enlarges in the time-history, however, is an unknown to be determined. A study on this question can be found in the paper by Chiriţă [5].…”

“…The spatial behavior of the transient solutions of the mathematical model based on the equations (1) and (2) was studied by Chiriţă [5] and Chiriţă et al [6], describing the depth of thermal penetration. In fact, it was established in [5] an influence domain result when m = n ± 1 , while for m = n there have been established some exponential decay estimates of the Saint-Venant type. These results have been established without requiring any restriction on the delay times other than the positivity of the product between the coefficients of the time derivatives of greatest order.…”