  Cahn-Hilliard-Navier-Stokes (CHNS) systems describes flows with two-phases,
e.g., a liquid with bubbles. Obtaining constitutive relations for general
dissipative processes for such a systems, which are thermodynamically
consistent, can be a challenge. We show how the metriplectic 4-bracket
formalism 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
thermodynamic conjugates of concentration and 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. This general CHNS system reduces to cases in the literature, to which
we can compare.
