From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

Gay–Balmaz, François; Yoshimura, Hidetoshi

doi:10.3390/e21010008

Cited by 48 publications

(60 citation statements)

References 59 publications

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“…The third paper of François Gay-Balmaz and Hiroaki Yoshimura [ 16 ] presents new results on the variational formulation of nonequilibrium thermodynamics for discrete or continuum systems, and its extension for irreversible processes. These new models are illustrated in the finite dimensional cases, and on the continuum side.…”

Section: Modern Fourier Heat Theorymentioning

confidence: 99%

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Barbaresco¹,

Gazeau

2019

Entropy

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For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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Section: Modern Fourier Heat Theorymentioning

confidence: 99%

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Barbaresco¹,

Gazeau

2019

Entropy

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“…The variational approach that we exploit in this paper is based on the Hamilton principle of classical mechanics and on its extension to include irreversible processes developed in [1,2,56], which is quickly reviewed below. Consider a mechanical system with configuration manifold Q and Lagrangian…”

Section: Introduction 1review Of the Literature Related To Porous Med...mentioning

confidence: 99%

Variational geometric approach to the thermodynamics of porous media

Gay–Balmaz¹,

Putkaradze²

2021

Preprint

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Many applications of porous media research involves high pressures and, correspondingly, exchange of thermal energy between the fluid and the matrix. While the system is relatively well understood for the case of non-moving porous media, the situation when the elastic matrix can move and deform, is much more complex. In this paper we derive the equations of motion for the dynamics of a deformable porous media which includes the effects of friction forces, stresses, and heat exchanges between the media, by using the new methodology of variational approach to thermodynamics [1,2]. This theory extends the recently developed variational derivation of the mechanics of deformable porous media [3] to include thermodynamic processes and can easily include incompressibility constraints. The model for the combined fluid-matrix system, written in the spatial frame, is developed by introducing mechanical and additional variables describing the thermal energy part of the system, writing the action principle for the system, and using a nonlinear, nonholonomic constraint on the system deduced from the second law of thermodynamics. The resulting equations give us the general version of possible friction forces incorporating thermodynamics, Darcy-like forces and friction forces similar to those used in the Navier-Stokes equations. The equations of motion are valid for arbitrary dependence of the kinetic and potential energies on the state variables. The results of our work are relevant for geophysical applications, industrial applications involving high pressures and temperatures, food processing industry, and other situations when both thermodynamics and mechanical considerations are important.

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“…Although significant efforts have been applied to the theoretical investigation of non-equilibrium thermodynamics, the theory has not reached the level of completeness due to the lack of a generalised variational formulation (Gay-Balmaz and Yoshimura, 2019). The modern approach to thermodynamics initiated in the twentieth century by Duhem in his work titled "Energetique" (Kondepudi and Prigogine, 2014), is directed towards the quantification of the "uncompensated transformation", which is the so-called "entropy production" by irreversible processes.…”

Section: Introductionmentioning

confidence: 99%

Formulation of steady-state void fraction through the principle of minimum entropy production

Giannetti

Saito

Yoshimura

2020

JTST

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The complexity of the actual operation of thermal engineering systems comprises multiphase interfacial phenomena evolving out of equilibrium. Therefore, their generalised formulation can contribute towards better understanding and control of these phenomena, eventually pushing the existing related technologies beyond the state-of-the-art. In this respect, variational principles are significant for a more comprehensive physical representation and for closing the problem, while obtaining relatively simpler mathematical formulations. In this study, a general variational formulation of dissipative two-phase flows based on the minimum entropy production is developed. In particular, this study provides a general expression of the entropy generation rate, which introduces interfacial contributions due to surface tension between different phases, and is used to estimate two-phase flow fraction based on Prigogine's theorem of minimum entropy generation. Subsequently, this formulation is investigated in terms of different assumptions and pressure drop models, and employed to clarify the implementation of Prigogine's theorem to obtain the widely-accepted Zivi's expression of void fraction and the effect of different assumptions on the deviation from his expression. A new expression is finally obtained to cover laminar flow conditions, which are implicitly excluded from the applicability of Zivi's expression.

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From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

Cited by 48 publications

References 59 publications

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Variational geometric approach to the thermodynamics of porous media

Formulation of steady-state void fraction through the principle of minimum entropy production

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