  We study local spin polarization in the relativistic hydrodynamic model.
Generalizing the Wigner functions previously obtained from chiral kinetic
theory by Y. Hidaka et al. [Phys. Rev. D 97, 016004 (2018)] to the massive
case, we present the possible contributions up to the order of $\hbar$ from
thermal vorticity, shear viscous tensor, other terms associated with the
temperature and chemical-potential gradients, and electromagnetic fields to the
local spin polarization. We then implement the (3+1)-dimensional viscous
hydrodynamic model to study the spin polarizations from these sources with a
small chemical potential and ignorance of electromagnetic fields by adopting an
equation of state different from those in other recent studies. Although the
shear correction alone upon the local polarization results in a sign and
azimuthal-angle dependence more consistent with experimental observations, as
also discovered in other recent studies, it is mostly suppressed by the
contributions from thermal vorticity and other terms that yield an opposite
trend. It is found that the total local spin polarization can be very sensitive
to the equation of states, the ratio of shear viscosity to entropy density, and
the freeze-out temperature.
