Bracket formulation for irreversible classical fields

Morrison, Philip

doi:10.1016/0375-9601(84)90635-2

Cited by 197 publications

(195 citation statements)

References 4 publications

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“…The Clebsch structure [10,11,12,13,14,15,16,17,18,19,20] differs from the Euler structure in the following points:…”

Section: Commentmentioning

confidence: 99%

“…We have chosen the state variables (14) because they appear to us as most appropriate. Another possible choice would be, for example, to replace the deformation gradient F with the displacement field.…”

Section: State Variablesmentioning

confidence: 99%

“…Our next task is to introduce equations governing the time evolution of the fields (14). We shall write them in the Cartesian Lagrangian coordinates y j .…”

Section: Time Evolutionmentioning

confidence: 99%

“…The state variables (14) represent the mesoscopic representative of the complete microscopic characterization that would consist of position and velocity coordinates of all microscopic particles composing the system. The stress tensor is a quantity representing interactions with exterior of the system, it is not a quantity characterizing states of the system.…”

Section: Stress-based Versus Strain-based Formulationmentioning

confidence: 99%

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Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Peshkov

Grmela

Romenski

2014

Continuum Mech. Thermodyn.

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On the example of two-phase continua experiencing stress induced solid-fluid phase transitions we explore the use of the Euler structure in the formulation of the governing equations. The Euler structure guarantees that solutions of the time evolution equations possessing it are compatible with mechanics and with thermodynamics. The former compatibility means that the equations are local conservation laws of the Godunov type and the latter compatibility means that the entropy does not decrease during the time evolution. In numerical illustrations, in which the one-dimensional Riemann problem is explored, we require that the Euler structure is also preserved in the discretization.

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“…The Clebsch structure [10,11,12,13,14,15,16,17,18,19,20] differs from the Euler structure in the following points:…”

Section: Commentmentioning

confidence: 99%

Section: State Variablesmentioning

confidence: 99%

“…Our next task is to introduce equations governing the time evolution of the fields (14). We shall write them in the Cartesian Lagrangian coordinates y j .…”

Section: Time Evolutionmentioning

confidence: 99%

Section: Stress-based Versus Strain-based Formulationmentioning

confidence: 99%

See 2 more Smart Citations

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Peshkov

Grmela

Romenski

2014

Continuum Mech. Thermodyn.

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“…5 of [2]). Kaufman [89], Morrison [90], and Grmela [66] published the first attempts to include dissipation in systems described by bracket formalisms by introducing a dissipation bracket. Grmela and Carreau [1], Grmela [38,91], and Beris and Edwards [2,67,68] used the bracket formalism to formulate the equations of change of polymeric liquids in a unified framework based on introducing the conformation tensor, an expectation value or continuum variable that represents local average values of stretch and orientation of the polymer coils.…”

Section: The Generalized Bracket Approach Of Grmela Beris and Edwardsmentioning

confidence: 99%

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

Pasquali

Scriven

2004

Journal of Non-Newtonian Fluid Mechanics

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A new method is presented for accounting for microstructural features of flowing complex fluids at the level of mesoscopic, or coarse-grained, models by ensuring compatibility with macroscopic and continuum thermodynamics and classical transport phenomena. In this method, the microscopic state of the liquid is described by variables that are local expectation values of microscopic features. The hypothesis of local thermodynamic equilibrium is extended to include information on the microscopic state, i.e., the energy of the liquid is assumed to depend on the entropy, specific volume, and microscopic variables. For compatibility with classical transport phenomena, the microscopic variables are taken to be extensive variables (per unit mass or volume), which obey convection-diffusion-generation equations. Restrictions on the constitutive laws of the diffusive fluxes and generation terms are derived by separating dissipation by transport (caused by gradients in the derivatives of the energy with respect to the state variables) and by relaxation (caused by non-equilibrated microscopic processes like polymer chain stretching and orientation), and by applying isotropy. When applied to unentangled, isothermal, non-diffusing polymer solutions, the equations developed according to the new method recover those developed by the Generalized Bracket [J. Non-Newtonian Fluid Mech. 23 (1987) Rheol. 38 (1994) 769]. Minor differences with published results obtained by the Generalized Bracket are found in the equations describing flow coupled to heat and mass transfer in polymer solutions. The new method is applied to entangled polymer solutions and melts in the general case where the rate of generation of entanglements depends nonlinearly on the rate of strain. A link is drawn between the mesoscopic transport equations of entanglements and conformation and the microscopic equation describing the configurational distribution of polymer segment stretch and orientation. Constraints are derived on the generation terms in the transport equations of entanglements and conformation, and the formula for the elastic stress is generalized to account for reversible formation and destruction of entanglements. A simplified version of the transport equation of conformation is presented which includes many previously published constitutive models, separates flow-induced polymer stretching and orientation, yet is simple enough to be useful for developing large-scale computer codes for modeling coupled fluid flow and transport phenomena in two-and three-dimensional domains with complex shapes and free surfaces.

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On the nonlinear stability and the existence of selective decay states of 3D quasi‐geostrophic potential vorticity equation

Esen

Han

Şengül

et al. 2019

Math Methods in App Sciences

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In this article, we study the dynamics of large scale motion in atmosphere and ocean governed by the 3D-quasi-geostrophic potential vorticity (QGPV) equation with a constant stratification. It is shown that for a Kolmogorov forcing on the first energy shell, there exist a family of exact solutions which are dissipative Rossby waves. The nonlinear stability of these exact solutions are analyzed based on the assumptions on the growth rate of the forcing. In the absence of forcing, we show the existence of selective decay states for the 3D-QGPV equation. The selective decay states are the 3D-Rossby waves traveling horizontally at a constant speed. All these results can be regarded as the expansion of that of the 2D-QGPV system and in the case of 3D-QGPV system with isotropic viscosity. Finally, we present a geometric foundation for the model as a general equation for non-equilibrium reversible-irreversible coupling.

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Bracket formulation for irreversible classical fields

Cited by 197 publications

References 4 publications

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Irreversible mechanics and thermodynamics of two-phase continua experiencing stress-induced solid–fluid transitions

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

On the nonlinear stability and the existence of selective decay states of 3D quasi‐geostrophic potential vorticity equation

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