  This paper presents a finite particle approximation of the two-fluid model
for liquid $^4$He 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 $^4$He 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 $^4$He.
