  In this paper, we present a detailed derivation of relativistic first-order
spin hydrodynamics using the Chapman-Enskog method to linearize the nonlocal
collision term for massive fermions proposed in \cite{Weickgenannt:2021cuo},
which well describes spin-orbit coupling in the collision process and is
relevant for the research on local spin polarization. Based on this collisional
term, we provide a formal discussion about first-order spin hydrodynamics and
determine the motion equations of fluid variables and nonequilibrium
corrections to the energy-momentum and spin tensors. The results indicate that
the motion equations show no differences compared to spinless first-order
hydrodynamics and the energy-momentum tensor receives no corrections from spin
as far as first-order theory is concerned, which calls for the construction of
second-order theory of fluids naturally incorporating the effect of spin-orbit
coupling.
