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

International audienceThe 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. However, real 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 systematically extended to include irreversible processes and nonequilibrium thermodynamics, through the new concept of thermodynamic displacement. Irreversible processes have also been 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. Unlike the variational formalism, most of these bracket formalisms do not follow from a systematic construction and have often been derived via a case by case approach, with slightly different axioms used in different situations. In this paper, we show that the variational formulation yields a constructive and systematic way to derive from a unified perspective these bracket formulations for fully compressible, multicomponent, multiphase fluids with a single temperature and velocity. In the case of a linear relation between the thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC brackets are recovered. Many previous results in the literature, typically obtained via heuristic approaches, are demonstrated to be special cases of this general formulation