Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Zaidni, Azeddine; Morrison, Philip J.; Benjelloun, Saad

doi:10.1016/j.physd.2024.134303

Thermodynamically consistent Cahn–Hilliard–Navier–Stokes equations using the metriplectic dynamics formalism

Journal Article · Wed Jul 24 00:00:00 EDT 2024 · Physica D: Nonlinear Phenomena

DOI:https://doi.org/10.1016/j.physd.2024.134303· OSTI ID:2998044

Zaidni, Azeddine ^[1]; ^[2]; Benjelloun, Saad ^[3]

Mohammed VI Polytechnic University, Ben Guerir (Morocco); University of Texas at Austin, TX (United States)
University of Texas at Austin, TX (United States)
Makhbar Mathematical Sciences Research Institute, Casablanca (Morocco); Léonard de Vinci Pôle Universitaire, Paris La Défense (France)

Cahn–Hilliard–Navier–Stokes (CHNS) systems describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such systems, which are thermodynamically consistent, can be a challenge. We show how the metriplectic 4-bracket formalism (Morrison and Updike, 2024) achieves this in a straightforward, in fact algorithmic, manner. First, from the noncanonical Hamiltonian formulation for the ideal part of a CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the concentration variable and its thermodynamics conjugate, the chemical potential, are included. Having expressions for the Hamiltonian (energy), entropy, and Poisson bracket, we describe a procedure for obtaining a metriplectic 4-bracket that describes thermodynamically consistent dissipative effects. The 4-bracket formalism leads naturally to a general CHNS system that allows for anisotropic surface energy effects. Furthermore, this general CHNS system reduces to cases in the literature, to which we can compare.

View Accepted Manuscript (DOE)

Research Organization:: The University of Texas at Austin, TX (United States)

Sponsoring Organization:: USDOE; USDOE Office of Science (SC), Fusion Energy Sciences (FES)

Grant/Contract Number:: FG02-04ER54742

OSTI ID:: 2998044

Alternate ID(s):: OSTI ID: 2426411

Journal Information:: Physica D: Nonlinear Phenomena, Journal Name: Physica D: Nonlinear Phenomena Vol. 468; ISSN 0167-2789

Publisher:: Elsevier BVCopyright Statement

Country of Publication:: United States

Language:: English

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