  Fermions become polarized in a vorticular fluid due to spin-vorticity
coupling. Such a polarization can be calculated from the Wigner function in a
quantum kinetic approach. Extending previous results for chiral fermions, we
derive the Wigner function for massive fermions up to the next-to-leading order
in spatial gradient expansion. The polarization density of fermions can be
calculated from the axial vector component of the Wigner function and is found
to be proportional to the local vorticity $\omega$. The polarizations per
particle for fermions and anti-fermions decrease with the chemical potential
and increase with energy (mass). Both quantities approach the asymptotic value
$\hbar\omega/4$ in the large energy (mass) limit. The polarization per particle
for fermions is always smaller than that for anti-fermions, whose ratio of
fermions to anti-fermions also decreases with the chemical potential. The
polarization per particle on the Cooper-Frye freeze-out hyper-surface can also
be formulated and is consistent with the previous result of Becattini et al..
