  Electron spin transport in a disordered metal is theoretically studied from
the hydrodynamic viewpoint focusing on the role of electron vorticity. The
spin-resolved momentum flux density of electrons is calculated taking account
of the spin-orbit interaction and uniform magnetization, and the expression for
the spin motive force is obtained as the linear response to a driving electric
field. It is shown that the spin-resolved momentum flux density and motive
force are characterized by troidal moments expressed as vector products of the
applied external electric field and the spin polarization and/or magnetization.
The spin-vorticity and magnetization-vorticity couplings studied recently are
shown to arise from the toridal moments contribution to the momentum flux
density. Spin motive force turns out to have a nonconservative contribution
besides the conventional conservative one due to the spin-vorticity coupling.
Spin accumulation induced by an electric field is calculated to demonstrate the
direct relation between vorticity and induced spin, and the spin Hall effect is
interpreted as due to the spin-vorticity coupling. The spin-vorticity coupling
is shown to give rise to a vorticity-induced torque and a spin relaxation. The
vorticity-induced torque is a linear effect of the spin-orbit interaction and
is expected to be larger than the second-order torques such as nonadiabatic
($\beta$) current-induced torque due to magnetization structure. The intrinsic
inverse spin Hall effect is argued to correspond to the antisymmetric
components of the momentum flux density in the hydrodynamic context.
