Representation of Viscous Dissipation in 2D Fluid Dynamics as a Symplectic Process and its Metriplectic Representation

TL;DR

This paper presents a novel symplectic and metriplectic framework for modeling viscous dissipation in 2D fluid dynamics, extending Hamiltonian mechanics to include dissipative effects through auxiliary variables.

Contribution

It introduces a new symplectic and metriplectic approach to represent viscous dissipation in non-canonical 2D fluid systems, incorporating entropy measures.

Findings

Derived a symplectic process for viscous dissipation.

Developed a metriplectic representation for the system.

Proposed an entropy measure for dissipative dynamics.

Abstract

Dissipation can be represented in Hamiltonian mechanics in an extended phase space as a symplectic process. The method uses an auxiliary variable which represents the excitation of unresolved dynamics and a Hamiltonian for the interaction between the resolved dynamics and the auxiliary variable. This method is applied to viscous dissipation (including hyper-viscosity) in a two-dimensional fluid, for which the dynamics is non-canonical. We derive a metriplectic representation and suggest a measure for the entropy of the system.