A spin-angular-momentum-conserving multiscale framework for reproducing macroscopic quantum phenomena in cryogenic liquid helium-4

We present a multiscale fluid dynamics framework for cryogenic liquid helium-4 based on a spin angular momentum conserving (SAMC) two-fluid model. The model describes helium as a mixture of inviscid and viscous components and formulates the viscous part using a modified Navier–Stokes equation that explicitly conserves spin angular momentum. This allows the treatment of rotating systems with dominant angular momentum effects. Unlike the Landau model, which focuses on condensate and thermal excitations, our approach considers helium as a continuum, enabling unified spatial-scale modeling of macroscopic flow phenomena. The SAMC-NS equations are derived under the nontrivial assumption of local non-rigid-body rotation. The internal angular velocity is introduced as a spin field, whose closure requires a physical interpretation that remains unclear. We resolve this by introducing a turbulence hierarchy, in which small-scale rotational flows are coarse-grained into rigid-body-like motion at larger scales. A convolution relation between small- and large-scale vorticity fields is established, leading to a recurrence formula for rotational viscosity across scales. Numerical simulations of decaying two-dimensional point-vortex turbulence confirm the upward transfer of vorticity under low viscosity. Finally, large-scale three-dimensional (3D) simulations using smoothed particle hydrodynamics on 256 GPUs demonstrate the scalability of the framework to fully 3D rotating helium systems.