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On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction

机译：时差双相滞后热模型的热力学一致性

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摘要

This paper deals with the time differential dual-phase-lag heat transfer models aiming, at first, to identify the eventually restrictions that make them thermodynamically consistent. At a first glance it can be observed that the capability of a time differential dual-phase-lag model of heat conduction to describe real phenomena depends on the properties of the differential operators involved in the related constitutive equation. In fact, the constitutive equation is viewed as an ordinary differential equation in terms of the heat flux components (or in terms of the temperature gradient) and it results that, for approximation orders greater than or equal to five, the corresponding characteristic equation has at least a complex root having a positive real part. That leads to a heat flux component (or temperature gradient) that grows to infinity when the time tends to infinity and so there occur some instabilities. Instead, when the approximation orders are lower than or equal to four, this is not the case and there is the need to study the compatibility with the Second Law of Thermodynamics. To this aim the related constitutive equation is reformulated within the system of the fading memory theory, and thus the heat flux vector is written in terms of the history of the temperature gradient and on this basis the compatibility of the model with the thermodynamical principles is analyzed.
机译：本文研究时差双相滞后传热模型，旨在首先确定使它们在热力学上保持一致的最终限制。乍一看，可以观察到热传导的时差双相滞后模型描述实际现象的能力取决于相关本构方程中所涉及的微分算子的性质。实际上，本构方程在热通量方面（或在温度梯度方面）被视为一个常微分方程，其结果是，对于近似阶数大于或等于5的情况，相应的特征方程具有至少具有正实部的复杂根。这导致热通量分量（或温度梯度）在时间趋于无穷大时增长到无穷大，因此会出现一些不稳定性。相反，当近似阶数小于或等于4时，情况并非如此，有必要研究与热力学第二定律的相容性。为此，在衰落记忆理论的系统内重新构造了相关的本构方程，从而根据温度梯度的历史记录了热通量矢量，并在此基础上分析了模型与热力学原理的兼容性。 。

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