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Entropy 2015, 17(3), 1329-1346; doi:10.3390/e17031329

Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation

Istituto dei Sistemi Complessi ISC-CNR, via Madonna del Piano 10, 50019 Sesto Fiorentino (Florence), Italy
Academic Editor: Ignazio Licata
Received: 15 November 2014 / Revised: 3 March 2015 / Accepted: 9 March 2015 / Published: 16 March 2015
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Abstract

The dynamics of dissipative fluids in Eulerian variables may be derived from an algebra of Leibniz brackets of observables, the metriplectic algebra, that extends the Poisson algebra of the frictionless limit of the system via a symmetric semidefinite component, encoding dissipative forces. The metriplectic algebra includes the conserved total Hamiltonian H, generating the non-dissipative part of dynamics, and the entropy S of those microscopic degrees of freedom draining energy irreversibly, which generates dissipation. This S is a Casimir invariant of the Poisson algebra to which the metriplectic algebra reduces in the frictionless limit. The role of S is as paramount as that of H, but this fact may be underestimated in the Eulerian formulation because S is not the only Casimir of the symplectic non-canonical part of the algebra. Instead, when the dynamics of the non-ideal fluid is written through the parcel variables of the Lagrangian formulation, the fact that entropy is symplectically invariant clearly appears to be related to its dependence on the microscopic degrees of freedom of the fluid, that are themselves in involution with the position and momentum of the parcel. View Full-Text
Keywords: fluid dynamics; Hamiltonian formulations; Lagrangian and Hamiltonian mechanics fluid dynamics; Hamiltonian formulations; Lagrangian and Hamiltonian mechanics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Materassi, M. Metriplectic Algebra for Dissipative Fluids in Lagrangian Formulation. Entropy 2015, 17, 1329-1346.

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