Least action principles for irreversible transport processes

Abstract

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 particle number of the given species. Lyapunov function reflects the irreversibility of a non-equilibrium transport process; we find that entropy production rate can serve as a Lyapunov function merely for an irreversible transport process governed by a constitutive relation in which the production of force and flux is equal to the entropy production rate, that is, the constitutive relation of entropy representation. It is noted that the constitutive relation of entropy representation cannot always coincide with the actual constitutive relation in practice. For example, heat conduction process is generally regarded as being governed by Fourier’s law; however, its constitutive relation of entropy representation can agree with the Fourier’s law only when the thermal conductivity is inversely proportional to the square of temperature. In fact, for an actual transport process, its Lyapunov function should be identical to its action that can derive the actual constitutive relation. It is the entransy dissipation rate that serves as both the action and Lyapunov function of thermal transport process. Lastly, we note that the combination of action and energy conservation equation cannot give the governing equation for a transient irreversible transport process; the quasi-least action principle can be obtained by introducing convolution integral.