Abstract

The Guyer–Krumhansl equation for the heat flux is a phenomenological bridge between Fourier heat transport (for size of the system much bigger than the mean free path of heat carriers) and hydrodynamic heat transport (for size of the system comparable to the mean free path of heat carriers). The corresponding phonon hydrodynamics is analogous to Newtonian hydrodynamics, but with the velocity replaced by the heat flux, the pressure gradient replaced by the temperature gradient and the shear viscosity replaced by the square of the mean-free path divided by the thermal conductivity. In this paper, we propose a nonlinear generalization of the Guyer–Krumhansl equation and phonon hydrodynamics based on an analogy with the power-law model of non-Newtonian fluids leading to a non-diffusive behaviour of heat transport. On the basis of this model, we obtain the corresponding nonlinear effective thermal conductivity of the model, depending on the radius of the channel and on the temperature gradient. The present proposal could be useful in the light of recent analyses of Poiseuille phonon hydrodynamics which suggest a non-Newtonian behaviour.

中文翻译：

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非线性声子流体动力学的幂律模型

摘要

热通量的 Guyer-Krumhansl 方程是傅立叶热传输（系统尺寸远大于热载体的平均自由程）和流体动力热传输（系统尺寸与平均自由程相当）之间的唯象桥梁。热载体）。相应的声子流体动力学类似于牛顿流体动力学，但速度由热通量代替，压力梯度由温度梯度代替，剪切粘度由平均自由程的平方除以热导率代替。在本文中，我们基于与非牛顿流体幂律模型的类比，提出了 Guyer-Krumhansl 方程和声子流体动力学的非线性推广，导致热传输的非扩散行为。在此模型的基础上，我们获得了模型相应的非线性有效导热系数，该导热系数取决于通道半径和温度梯度。鉴于最近对泊肃叶声子流体动力学的分析表明了非牛顿行为，本提议可能有用。