标题： 热传导的速率方程建模 显示英文标题

标题： Modeling of heat conduction through rate equations

摘要： 从经典的热力学方法出发，我们推导出速率型方程来描述可变形介质中的热量流动行为。 本构方程在物质（拉格朗日）描述中定义，其中标准时间导数满足客观性原理。 第二定律的表述以经典形式给出，然后按照Coleman-Noll方法发展热力学限制条件。 然而，我们考虑的是相应的等式而不是Clausius Duhem不等式，其中熵产率由一个非负的本构函数规定。 自由能和熵产率都被假设依赖于一组共同的独立变量，除了温度外，还包括温度梯度、热通量矢量以及它们的时间导数。 这种方法导致了与第二定律内在一致且易于分析的热通量矢量的速率型本构方程。 除了获得已知的模型（例如，Cattaneo-Maxwell模型、Jeffreys类似模型和Green-Naghdi热导体），该方案还允许我们建立新的和更复杂的热传输模型，这些模型可能在描述纳米系统中的热行为方面有应用。 事实上，当涉及热通量矢量的高阶时间导数时，速率方程中会出现许多不同的松弛时间。 显示英文摘要

摘要： Starting from a classical thermodynamic approach, we derive rate-type equations to describe the behavior of heat flow in deformable media. Constitutive equations are defined in the material (Lagrangian) description where the standard time derivative satisfies the principle of objectivity. The statement of the Second Law is formulated in the classical form and the thermodynamic restrictions are then developed following the Coleman-Noll procedure. However, instead of the Clausius Duhem inequality we consider the corresponding equality where the entropy production rate is prescribed by a non-negative constitutive function. Both the free energy and the entropy production are assumed to depend on a common set of independent variables involving, in addition to temperature, both temperature gradient and heat-flux vector together with their time derivatives. This approach results in rate-type constitutive equations for the heat-flux vector that are intrinsically consistent with the Second Law and easily amenable to analysis. In addition to obtaining already known models (e.g. Cattaneo-Maxwell's, Jeffreys-like and Green-Naghdi's heat conductors), this scheme allows us to build new and more complex models of heat transport that may have applications in describing the thermal behavior in nanosystems. Indeed, when higher order time derivatives of the heat flux vector are involved, many different relaxation times occur within the rate equation.