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How to properly describe continuum thermodynamics of binary mixtures whereeach constituent has its own momentum? The Symmetric HyperbolicThermodynamically Consistent (SHTC) framework and Hamiltonian mechanics in theform of the General Equation for Non-Equilibrium Reversible-IrreversibleCoupling (GENERIC) provide two answers, which are similar but not identical,and are compared in this article. They are compared both analytically andnumerically on several levels of description, varying in the amount of detail.The GENERIC equations, stemming from the Liouville equation, contain termsexpressing self-advection of the relative velocity by itself, which lead to avorticity-dependent diffusion matrix after a reduction. The SHTC equations, onthe other hand, do not contain such terms. We also show how to formulate atheory of mixtures with two momenta and only one temperature that is compatiblewith the Liouville equation and possesses the Hamiltonian structure, includingJacobi identity.

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