Retained-spin micropolar hydrodynamics from the Boltzmann--Curtiss equation: a generalized Chapman--Enskog construction

Abstract

We derive a retained-spin micropolar hydrodynamic closure from the Boltzmann--Curtiss equation using a generalized Chapman--Enskog construction in which the local mean spin is retained as a quasi-slow variable. Starting from the exact kinetic balance laws for mass, linear momentum, and intrinsic angular momentum, we isolate the bookkeeping relation between antisymmetric stress and stress-induced spin torque, decompose the first-order source into irreducible scalar, axial, and symmetric-traceless sectors, and show explicitly how the standard micropolar constitutive structure with coefficients $(η,ξ,η_r,α,β,γ)$ emerges. This decomposition makes clear that the one-particle kinetic stress contributes only to the symmetric stress, whereas the rotational viscosity belongs to an intrinsic/collisional transfer channel. For perfectly rough elastic hard spheres, we further obtain explicit dilute-gas estimates for the rotational viscosity $η_r$ from homogeneous spin relaxation and for the transverse spin-diffusion combination $β+γ$ from a transport-relaxation calculation. Targeted event-driven molecular-dynamics simulations are used as a posteriori checks: expanded homogeneous-spin density and roughness sweeps support the predicted $n^2$ and $K/(K+1)$ trends for $η_r$, while finite-$k$ transverse runs provide a qualitative diagnostic of the retained-spin response. The result is a self-contained derivation and coefficient-level estimate of retained-spin micropolar hydrodynamics that clarifies which parts of the closure are exact, which are first-order generalized Chapman--Enskog results, and which remain controlled rough-sphere estimates.

Figures (4)

• Figure 1: Homogeneous spin relaxation in EDMD for perfectly rough elastic hard spheres at $N=8192$, $\phi=0.020$, and $K=0.400$. The solid curve is the ensemble-averaged magnitude of the mean spin, and the dashed line is the single-exponential fit used to extract $\nu_{\mathrm{spin}}$ and hence $\eta_r$ via Eq. \ref{['eq:edmd_homogeneous_fit']}. The shaded interval marks the fit window.
• Figure 2: Rotational viscosity extracted from homogeneous-spin EDMD runs as a function of $n^2$ at fixed $K=0.400$ for the expanded density sweep $0.005\le\phi\le0.050$. The dashed line is a guide proportional to $n^2$, normalized through the better-conditioned points ($R^2_{\log}\ge 0.90$).
• Figure 3: Rotational viscosity extracted from homogeneous-spin EDMD runs as a function of the reduced moment-of-inertia parameter $K$ at fixed $\phi=0.020$ for the expanded roughness sweep $0.050\le K\le1.000$. The dashed curve is a guide proportional to $K/(K+1)$, normalized through the better-conditioned points ($R^2_{\log}\ge 0.90$).
• Figure 4: Representative finite-$k$ retained-spin EDMD run at $N=8192$, $\phi=0.020$, $K=0.400$, and $k=0.14996$ (batch B2). The solid curves are the ensemble-averaged complex Fourier amplitudes and the dashed curves are the best-fit linear two-field model from Eq. \ref{['eq:edmd_transverse_matrix']}. The $\hat{\zeta}_z$ component is reproduced reasonably well, but the $\hat{\omega}_z$ component remains noisy and the off-diagonal extractions of $\eta_r$ are not mutually consistent. In the present data this figure should therefore be interpreted as a qualitative diagnostic rather than as a high-precision coefficient measurement.