A Variational Principle for Dissipative Fluid Dynamics

Fukagawa, Hiroki; Fujitani, Youhei

doi:10.1143/ptp.127.921

Cited by 28 publications

(51 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There have been a number of (more or less successful) attempts to make progress on building dissipative variational models. A common approach has been to combine a variational model for the non-dissipative aspects with an argument that constrains the entropy production, often involving Lagrange multipliers (see Ichiyanagi 1994 for a review and Djukic and Vujanovic 1975;Djukic and Strauss 1980;Mobbs 1982;Kobe et al 1986;Vujanovic et al 1986;Honein et al 1991;Chien et al 1996;Nordbrock and Kienzler 2007;Fukagawa and Fujitani 2012 for samples of the literature). The model we will consider is conceptually different.…”

Section: Towards a Dissipative Action Principlementioning

confidence: 99%

Relativistic fluid dynamics: physics for many different scales

Andersson

Comer

2021

Living Rev Relativ

View full text Add to dashboard Cite

The relativistic fluid is a highly successful model used to describe the dynamics of many-particle systems moving at high velocities and/or in strong gravity. It takes as input physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process—e.g., drawing on astrophysical observations—an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as colliding heavy ions in laboratory experiments, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multi-) fluid model. We focus on the variational principle approach championed by Brandon Carter and collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory. The formalism provides a foundation for complex models, e.g., including electromagnetism, superfluidity and elasticity—all of which are relevant for state of the art neutron-star modelling.

show abstract

Section: Towards a Dissipative Action Principlementioning

confidence: 99%

Relativistic fluid dynamics: physics for many different scales

Andersson

Comer

2021

Living Rev Relativ

View full text Add to dashboard Cite

show abstract

“…Remark 3.8. Another variational approach to the Navier-Stokes-Fourier system has been developed by Fukagawa and Fujitani [2012], in which the internal conversion due to frictional forces from the mechanical to the heat power in dissipated systems is written as nonholonomic constraints in the integral form over the fluid domain. However, this variational approach does not involve the variables Γ nor Σ.…”

Section: Spatial Representationmentioning

confidence: 99%

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems

Gay–Balmaz

Yoshimura

2017

Journal of Geometry and Physics

138

View full text Add to dashboard Cite

Part I of this paper introduced a Lagrangian variational formulation for the nonequilibrium thermodynamics of discrete systems. This variational formulation extends the Hamilton principle to allow the inclusion of irreversible processes in the dynamics. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of entropy production associated to all the irreversible processes involved. In Part II, we develop this formulation for the case of continuum systems by extending the setting of Part I to infinite dimensional nonholonomic Lagrangian systems. The variational formulation is naturally expressed in material representation, while its spatial version is obtained via a nonholonomic Lagrangian reduction by symmetry. The theory is illustrated with the examples of a viscous heat conducting fluid and its multicomponent extension including chemical reactions and mass transfer.

show abstract

“…More recently, Fukagawa and Fujitani [2012] showed a variational formulation of viscoelastic fluids, in which the internal conversion of mechanical power into heat power due to frictional forces was written as a nonholonomic constraint.…”

Section: Introductionmentioning

confidence: 99%

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

Gay–Balmaz

Yoshimura

2017

Journal of Geometry and Physics

120

View full text Add to dashboard Cite

In this paper, we present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation is an extension of the Hamilton principle in classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of entropy production associated to all the irreversible processes involved. Hence from a mathematical point of view, our variational formulation may be regarded as a generalization of the Lagrange-d'Alembert principle used in nonlinear nonholonomic mechanics to the nonequilibrium thermodynamics, where the conventional Lagrange-d'Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated variational constraint must be separately treated. In our approach, to deal with the nonlinear nonholonomic constraint, we introduce a variable called the thermodynamic displacement associated to each irreversible process. This allows us systematically to define the corresponding variational constraint. In Part I, our variational theory is illustrated with various examples of discrete systems such as mechanical systems with friction, matter transfer, electric circuits, chemical reactions, and diffusion across membranes. In Part II of the present paper, we will extend our variational theory of discrete systems to the case of of continuum systems. Contents 1 Introduction 2 2 Some preliminaries 5 3 Variational formulation for nonequilibrium thermodynamics of simple systems 9 3.

show abstract

A Variational Principle for Dissipative Fluid Dynamics

Cited by 28 publications

References 13 publications

Relativistic fluid dynamics: physics for many different scales

Relativistic fluid dynamics: physics for many different scales

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part II: Continuum systems

A Lagrangian variational formulation for nonequilibrium thermodynamics. Part I: Discrete systems

Contact Info

Product

Resources

About