Retained-spin micropolar hydrodynamics from the Boltzmann--Curtiss equation

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 one-particle kinetic balance identities and the corresponding exact coarse-grained finite-size balances for mass, linear momentum, and intrinsic angular momentum, we keep the collisional-transfer contribution to the antisymmetric stress explicit in the spin balance, decompose the first-order source into irreducible scalar, axial, and symmetric-traceless sectors, and show explicitly how the standard micropolar constitutive structure with coefficients $(\eta,\xi,\eta_r,\alpha,\beta,\gamma)$ emerges. This decomposition makes clear that the one-particle kinetic stress contributes only to the symmetric stress, whereas the rotational viscosity belongs to a collisional-transfer channel. For perfectly rough elastic hard spheres, we further obtain explicit dilute-gas estimates for the rotational viscosity $\eta_r$ from homogeneous spin relaxation and for the transverse spin-diffusion combination $\beta+\gamma$ 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 $\eta_r$, 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 balance-law statements, which are first-order generalized Chapman--Enskog results, and which remain controlled rough-sphere estimates.