Abstract
This paper presents a finite particle approximation of the two-fluid model for liquid 4He using smoothed particle hydrodynamics (SPH). In recent years, several studies have combined the vortex filament model (VFM), which describes quantized vortices in superfluid components, with the Navier–Stokes equations, which describe the motion of normal fluids. These studies led us to assume that coupling both components of the two-fluid model instead of using the VFM to describe the superfluid component enables us to approximate the system. In this study, we formulated a new SPH model that simultaneously solves both equations of motion of the two-fluid model. We then performed a numerical simulation of the rotating liquid 4He using our SPH. The results showed that the two major phenomena, the emergence of multiple independent vortices parallel to the circular axis and that of the so-called rigid-body rotation, can be reproduced by solving the two-fluid model using SPH. This finding is interesting because it was previously assumed that only a single vortex emerges when addressing similar problems without considering quantum mechanics. Our further analysis found that the emergence of multiple independent vortices can be realized by reformulating the viscosity term of the two-fluid model to conserve the angular momentum of the particles around their axes. Consequently, our model succeeded in reproducing the phenomena observed in quantum cases, even though we solve the phenomenological governing equations of liquid 4He.
Highlights
The bizarre behavior of liquid helium 4 has attracted the attention of condensed matter physicists for many years; the detailed dynamics of this fluid remain a mystery even eighty years after its discovery.Figure 1 presents an overview of the basic properties of 4He in the form of a schematic phase diagram
We propose to discretize the two-fluid model of superfluid 4He using smoothed particle hydrodynamics (SPH) [18], a well-established Lagrangian particle approximation that is popular in the field of astrophysics
To solve the governing equations, we introduced an auxiliary equation representing the microscopic relationship between entropy and temperature, which was derived from quantum statistical mechanics in elementary particle physics
Summary
The bizarre behavior of liquid helium 4 (hereinafter, 4He) has attracted the attention of condensed matter physicists for many years; the detailed dynamics of this fluid remain a mystery even eighty years after its discovery. On the premise that this combined approach is acceptable, we can assume that coupling both components of the two-fluid model, instead of using the VFM for the superfluid component, would approximate the system This approximation is allowed only when we use the twofluid model to directly simulate large-scale problems on the continuum mechanical scale because technically, this breaks the microscopic laws of quantum mechanics. Scenarios such as these, which simultaneously solve for the two components of the two-fluid model, correspond to multi-phase flows in a classical fluid.
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