Hydrodynamics of superfluid 4He without dissipative effects

This review paper is the first of a series of papers focusing on the singular behavior of superfluids. Here, we will consider the laminar flow of superfluid $^4$He. It is shown that the properties of helium II can be explained both considering it as a two-fluid mixture or as a single fluid with extremely high thermal conductivity and extremely small viscosity. More specifically, in this paper is shown that the anomalous effects in helium II are, in a large measure, a consequence of entropy conservation. Indeed, it will be shown that these effects can be explained imposing entropy conservation in a two-fluid mixture as well as in a single fluid described by extended thermodynamics. Firstly, the rational theory of mixtures of ideal fluids is applied to describe the anomalous behaviour of superfluid helium II. Then, rational extended thermodynamics is applied to a fluid where entropy is conserved. The paper is not exclusively a review of already published works, but it contains several new results and proposals which are formulated in it for the first time.