不可逆输运过程的最小作用量原理  

Least action principles for irreversible transport processes

作　　者：华钰超 过增元[1] Yuchao Hua;Zengyuan Guo(Key Laboratory for Thermal Science and Power Engineering of Ministry of Education,Department of Engineering Mechanics,Tsinghua University,Beijing 100084,China)

机构地区：[1]清华大学工程力学系,热科学与动力工程教育部重点实验室,北京100084

出　　处：《科学通报》2021年第2期253-260,共8页Chinese Science Bulletin

基　　金：国家自然科学基金(51906121,51825601,51676108,51321002);博士后创新人才计划(BX20180155);中国博士后科学基金(2018M641348)资助。

摘　　要：最小作用量原理最早应用于光学和力学等可逆的物理过程,其后人们基于熵产发展了不同表述的最小作用量原理来处理不可逆输运过程,昂萨格最小能量耗散原理就是其中的代表.然而,基于熵产的最小作用量原理并不一定对应符合实际情况的本构关系(例如不能导出傅里叶导热定律).本文通过分析表明,导热过程的最小作用量不是熵产率而是(与热质能成正比)耗散率.我们对质量和动量不可逆输运过程的最小作用量原理进行了探讨,发现不可逆定态线性输运过程的作用量应该由真实力与流的点积得到的耗散率与瑞利函数进行构造,它们都可以看作某种能量的耗散率.此外,还发现了符合实际本构关系的不可逆线性输运过程的李雅普诺夫函数并不是熵产率,不可逆输运过程的作用量和李雅普诺夫函数应该是统一的,即都是某种能量的耗散率.分析还表明,对于瞬态不可逆输运过程,定态条件下的作用量与能量守恒方程的结合并不能导出相应的控制方程;需要使用卷积积分构建的准作用量才能导出瞬态控制方程.Least action principles(i.e., variational principles) were firstly applied to describe reversible processes, such as geometrical optics and mechanics. Then, they were extended to handle irreversible processes on the basis of entropy production rate,and the principle of least dissipation of energy by Onsager is one of the representative theorems. Nevertheless, it should be noted that the least action principles based on entropy production rate cannot always derive the actual constitutive relation of an irreversible transport process. As an example, the least action principles based on entropy production rate, including the principle of least dissipation of energy, will not derive the Fourier’s heat conduction law that is the fundamental constitutive relation of thermal transport process. In order to resolve this problem, Guo et al. proposed a new quantity,entransy dissipation rate, which is defined as the production of heat flux and negative temperature gradient and proportional to the dissipation rate of the potential energy of thermomass. It has been proven that the action of heat conduction should be the entransy dissipation rate;moreover, a theoretical framework for heat transfer optimization has been built based on it. In the present work, we extend the entransy theory to deal with various irreversible transport processes, including mass and momentum transport, and analyze their least action principles and Lyapunov functions. It is found that the action for a specific irreversible transport process can be constructed using the Rayleigh dissipation function and the product of actual driving force and flux, which can be regarded as the dissipation of some type of energy. For instance, the action for the momentum transport process governed by Newton’s viscosity law can be constructed using the dissipation of mechanic energy, while the action for mass transport process described by Fick’s law is related to the dissipation of chemical potential that is the energy absorbed or released due to a change of the p

关 键 词：最小作用量原理 不可逆线性输运过程 变分 李雅普诺夫函数 理论 

分 类 号：O4-0[理学—物理]