Single and double generator bracket formulations of multicomponent fluids with irreversible processes

Abstract

The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known variational formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian formulation that involves a Poisson bracket structure. These variational and bracket structures underlie many of the most basic principles that we know about geophysical fluid flows, such as conservation laws. However, real geophysical flows also include irreversible processes, such as viscous dissipation, heat conduction, diffusion and phase changes. Recent work has demonstrated that the variational formulation can be extended to include irreversible processes and non-equilibrium thermodynamics, through the new concept of thermodynamic displacement. By design, and in accordance with fundamental physical principles, the resulting equations automatically satisfy the first and second law of thermodynamics. Irreversible processes can also be incorporated into the bracket structure through the addition of a dissipation bracket. This gives what are known as the single and double generator bracket formulations, which are the natural generalizations of the Hamiltonian formulation to include irreversible dynamics. Here the variational formulation for irreversible processes is shown to underlie these bracket formulations for fully compressible, multicomponent, multiphase geophysical fluids with a single temperature and velocity. Many previous results in the literature are demonstrated to be special cases of this approach. Finally, some limitations of the current approach (especially with regards to precipitation and nonlocal processes such as convection) are discussed, and future directions of research to overcome them are outlined.

From Lagrangian Mechanics to Nonequilibrium Thermodynamics: A Variational Perspective

“…This was further developed, for instance, in [ 31 , 32 , 33 , 34 , 35 ]. A systematic construction of such brackets from the variational formulation given in the present paper was presented in [ 36 ] for the thermodynamics of multicomponent fluids.…”

A general metriplectic framework with application to dissipative extended magnetohydrodynamics

“…Less known is the fact that dissipative dynamics can also emerge from brackets [7,8,9,10,11,12] and the entropy S rather than the Hamiltonian H may serve as the generating function. (See [13,14,15,16,17,18,19,20,21] for a selection of more recent theoretical work, and e.g. [22,23,24] for recent numerical algorithms based on bracket dissipative structure.)…”

“…They are also known in other subjects of physics, e.g., in geophysical fluids [29] and elasticity [30]. Fewer metriplectic brackets are known; however, they have been discovered for fluids with viscosity and thermal diffusion [10], n-fluid models with chemical reactions [20], elasticity [30], and magnetohydrodynamics [31].…”

“…Less known is the fact that dissipative dynamics can also emerge from brackets (Kaufman & Morrison 1982;Grmela 1984;Kaufman 1984;Morrison 1984aMorrison ,b, 1986 and the entropy S rather than the Hamiltonian H may serve as the generating function. (See Grmela & Öttinger (1997a,b), Edwards (1998), Morrison (1998b), Kimura & Morrison (2014), Gay-Balmaz & Yoshimura (2017a,b), Eldred & Gay-Balmaz (2018) and Materassi & Morrison (2018) for a selection of more recent theoretical work, and e.g. Kraus & Hirvijoki (2017), Morrison (2017) and Bressan et al (2018) for recent numerical algorithms based on bracket dissipative structure.)…”

“…Fewer metriplectic brackets are known; however, they have been discovered for fluids with viscosity and thermal diffusion (Morrison 1984a), elasticity (Edwards & Beris 1991) and magnetohydrodynamics (Materassi & Tassi 2012). A comprehensive Lagrangian-based approach, as opposed to our bracket approach, is given for n-fluid models with chemical reactions and general multicomponent fluids with irreversible processes in Eldred & Gay-Balmaz (2018, 2020.…”