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

The Cahn-Hilliard-Navier-Stokes (CHNS) equations describe flows with two-phases, e.g., a liquid with bubbles. Obtaining constitutive relations for general dissipative processes for such a system, which are thermodynamically consistent, can be a challenge. We show how the metriplectic formalism of [1-4] achieves this in a straightforward manner. First, from the noncanonical Hamiltonian formulation for the ideal part of the CHNS system we obtain an appropriate Casimir to serve as the entropy in the metriplectic formalism [1-3] that describes the dissipation (e.g. viscosity, heat conductivity and diffusion effects). General thermodynamics with the thermodynamic conjugates of concentration and chemical potential are included. The formulation leads naturally to the more general case that allows for anisotropic surface energy effects.

[1] P. J. Morrison & M. Updike, arXiv:2306.06787v1 [math-ph] 11 Jun 2023.

[2] P. J. Morrison, Phys. Lett. A 100, 423 (1984).

[3] P. J. Morrison, Tech. Rep. PAM--228, Univ. Cal. Berkeley (1984).

[4] P. J. Morrison, Physica D 18, 410 (1986).