Superfluid helium-4 is characterized by extremely small values of kinematic viscosity, and its thermal conductivity can be huge, orders of magnitude larger than that of water or air. Additionally, quantum vortices may exist within the fluid. Therefore, its behavior cannot be explained by using the classical tools of Newtonian fluid mechanics, and, over the years, a few alternative models have been proposed. In order to highlight similarities and differences between these models, we recast them within a unifying framework, the general equation for non-equilibrium reversible-irreversible coupling (GENERIC). We begin by comparing the original two-fluid model, developed by Tisza and Landau, with the Hall–Vinen–Bekarevich–Khalatnikov model, both prescribing two types of fluid motion and two fluid densities, at flow scales appreciably larger than the typical distance between quantum vortices. We find from the geometrical structure of the models that only one fluid density plays the role of state variable, which should be taken into account when choosing an adequate expression for the free energy. We also recast within the GENERIC framework the one-fluid model of superfluid helium-4, where the inviscid component of two-fluid models is replaced by a caloric quantity, such as entropy. We find that the corresponding geometrical structures are analogous, with the roles of density and entropy swapped. In short, our work demonstrates that the studied models are compatible with each other, at least when focusing on the reversible parts of the models.
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Note that calling field a velocity might be problematic in a non-Euclidean space, where vectors and covectors must be distinguished, because is actually a covector field (like momentum), not a vector field (like velocity).
77.
Note that the pressure loses its thermodynamic interpretation in the incompressible case and it is reconstructed from the requirement that the velocity fields be divergence-free.42
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Note that the fraction on the right hand side is interpreted component by component, since the quantities in the numerator and denominator are collinear in the case of isotropic e0.
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