Comparison of the Symmetric Hyperbolic Thermodynamically Compatible framework with Hamiltonian mechanics of binary mixtures

TL;DR

This paper compares the Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework with Hamiltonian mechanics (GENERIC) for describing the thermodynamics of binary mixtures, analyzing their differences and similarities analytically and numerically.

Contribution

It provides a detailed comparison of SHTC and GENERIC frameworks, highlighting their differences in modeling binary mixtures with multiple momenta.

Findings

GENERIC includes self-advection terms leading to vorticity-dependent diffusion.

SHTC does not contain self-advection terms.

A formulation of mixtures with two momenta and one temperature compatible with Liouville and Hamiltonian structures is proposed.

Abstract

How to properly describe continuum thermodynamics of binary mixtures where each constituent has its own momentum? The Symmetric Hyperbolic Thermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in the form of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) provide two answers, which are similar but not identical, and are compared in this article. They are compared both analytically and numerically on several levels of description, varying in the amount of detail. The GENERIC equations, stemming from the Liouville equation, contain terms expressing self-advection of the relative velocity by itself, which lead to a vorticity-dependent diffusion matrix after a reduction. The SHTC equations, on the other hand, do not contain such terms. We also show how to formulate a theory of mixtures with two momenta and only one temperature that is compatible with the Liouville equation and possesses the Hamiltonian structure, including Jacobi identity.